question_id string | subfield string | context string | question string | images images list | final_answer list | is_multiple_answer bool | unit string | answer_type string | error string | source string |
|---|---|---|---|---|---|---|---|---|---|---|
1735 | Geometry | null | Three circular arcs $\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ connect the points $A$ and $C$. These arcs lie in the same half-plane defined by line $A C$ in such a way that $\operatorname{arc} \gamma_{2}$ lies between the $\operatorname{arcs} \gamma_{1}$ and $\gamma_{3}$. Point $B$ lies on the segment $A C$. Let $h_{1... | null | true | null | null | null | TP_MM_maths_en_COMP | |
1975 | Combinatorics | null | Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them Sand Z-tetrominoes, respectively.
S-tetrominoes... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2039 | Combinatorics | null | An anti-Pascal pyramid is a finite set of numbers, placed in a triangle-shaped array so that the first row of the array contains one number, the second row contains two numbers, the third row contains three numbers and so on; and, except for the numbers in the bottom row, each number equals the absolute value of the di... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2184 | Geometry | null | Let $A B C D$ be a cyclic quadrilateral, and let diagonals $A C$ and $B D$ intersect at $X$. Let $C_{1}, D_{1}$ and $M$ be the midpoints of segments $C X$, $D X$ and $C D$, respectively. Lines $A D_{1}$ and $B C_{1}$ intersect at $Y$, and line $M Y$ intersects diagonals $A C$ and $B D$ at different points $E$ and $F$, ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2227 | Geometry | null | Let $H$ be the orthocenter and $G$ be the centroid of acute-angled triangle $\triangle A B C$ with $A B \neq A C$. The line $A G$ intersects the circumcircle of $\triangle A B C$ at $A$ and $P$. Let $P^{\prime}$ be the reflection of $P$ in the line $B C$. Prove that $\angle C A B=60^{\circ}$ if and only if $H G=G P^{\p... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2270 | Geometry | null | In the diagram, two circles are tangent to each other at point $B$. A straight line is drawn through $B$ cutting the two circles at $A$ and $C$, as shown. Tangent lines are drawn to the circles at $A$ and $C$. Prove that these two tangent lines are parallel.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2274 | Combinatorics | null | A school has a row of $n$ open lockers, numbered 1 through $n$. After arriving at school one day, Josephine starts at the beginning of the row and closes every second locker until reaching the end of the row, as shown in the example below. Then on her way back, she closes every second locker that is still open. She con... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2348 | Geometry | null | In the diagram, $C$ lies on $B D$. Also, $\triangle A B C$ and $\triangle E C D$ are equilateral triangles. If $M$ is the midpoint of $B E$ and $N$ is the midpoint of $A D$, prove that $\triangle M N C$ is equilateral.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2363 | Geometry | null | In parallelogram $A B C D, A B=a$ and $B C=b$, where $a>b$. The points of intersection of the angle bisectors are the vertices of quadrilateral $P Q R S$.
Prove that $P Q R S$ is a rectangle. | null | true | null | null | null | TP_MM_maths_en_COMP | |
2364 | Geometry | null | In parallelogram $A B C D, A B=a$ and $B C=b$, where $a>b$. The points of intersection of the angle bisectors are the vertices of quadrilateral $P Q R S$.
Prove that $P R=a-b$. | null | true | null | null | null | TP_MM_maths_en_COMP | |
2367 | Geometry | null | An equilateral triangle $A B C$ has side length 2 . A square, $P Q R S$, is such that $P$ lies on $A B, Q$ lies on $B C$, and $R$ and $S$ lie on $A C$ as shown. The points $P, Q, R$, and $S$ move so that $P, Q$ and $R$ always remain on the sides of the triangle and $S$ moves from $A C$ to $A B$ through the interior of ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2398 | Geometry | null | In the diagram, line segment $F C G$ passes through vertex $C$ of square $A B C D$, with $F$ lying on $A B$ extended and $G$ lying on $A D$ extended. Prove that $\frac{1}{A B}=\frac{1}{A F}+\frac{1}{A G}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2448 | Geometry | null | A circle with its centre on the $y$-axis intersects the graph of $y=|x|$ at the origin, $O$, and exactly two other distinct points, $A$ and $B$, as shown. Prove that the ratio of the area of triangle $A B O$ to the area of the circle is always $1: \pi$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2449 | Geometry | null | In the diagram, triangle $A B C$ has a right angle at $B$ and $M$ is the midpoint of $B C$. A circle is drawn using $B C$ as its diameter. $P$ is the point of intersection of the circle with $A C$. The tangent to the circle at $P$ cuts $A B$ at $Q$. Prove that $Q M$ is parallel to $A C$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2462 | Geometry | null | A large square $A B C D$ is drawn, with a second smaller square $P Q R S$ completely inside it so that the squares do not touch. Line segments $A P, B Q, C R$, and $D S$ are drawn, dividing the region between the squares into four nonoverlapping convex quadrilaterals, as shown. If the sides of $P Q R S$ are not paralle... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2474 | Geometry | null | In triangle $A B C, \angle A B C=90^{\circ}$. Rectangle $D E F G$ is inscribed in $\triangle A B C$, as shown. Squares $J K G H$ and $M L F N$ are inscribed in $\triangle A G D$ and $\triangle C F E$, respectively. If the side length of $J H G K$ is $v$, the side length of $M L F N$ is $w$, and $D G=u$, prove that $u=v... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2488 | Geometry | null | In the diagram, quadrilateral $A B C D$ has points $M$ and $N$ on $A B$ and $D C$, respectively, with $\frac{A M}{A B}=\frac{N C}{D C}$. Line segments $A N$ and $D M$ intersect at $P$, while $B N$ and $C M$ intersect at $Q$. Prove that the area of quadrilateral $P M Q N$ equals the sum of the areas of $\triangle A P D$... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2499 | Geometry | null | In the diagram, $A B$ and $B C$ are chords of the circle with $A B<B C$. If $D$ is the point on the circle such that $A D$ is perpendicular to $B C$ and $E$ is the point on the circle such that $D E$ is parallel to $B C$, carefully prove, explaining all steps, that $\angle E A C+\angle A B C=90^{\circ}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2516 | Number Theory | null | Suppose that $m$ and $n$ are positive integers with $m \geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is
The $(3,4)$-sawtooth sequence in... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2526 | Geometry | null | In the diagram, $A B C D$ is a square. Points $E$ and $F$ are chosen on $A C$ so that $\angle E D F=45^{\circ}$. If $A E=x, E F=y$, and $F C=z$, prove that $y^{2}=x^{2}+z^{2}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2538 | Geometry | null | In the diagram, $A B$ is tangent to the circle with centre $O$ and radius $r$. The length of $A B$ is $p$. Point $C$ is on the circle and $D$ is inside the circle so that $B C D$ is a straight line, as shown. If $B C=C D=D O=q$, prove that $q^{2}+r^{2}=p^{2}$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2542 | Algebra | null | Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown below.
Throughout this problem,... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2547 | Geometry | null | In trapezoid $A B C D, B C$ is parallel to $A D$ and $B C$ is perpendicular to $A B$. Also, the lengths of $A D, A B$ and $B C$, in that order, form a geometric sequence. Prove that $A C$ is perpendicular to $B D$.
(A geometric sequence is a sequence in which each term after the first is obtained from the previous ter... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2561 | Geometry | null | In the graph, the parabola $y=x^{2}$ has been translated to the position shown.
Prove that $d e=f$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2562 | Geometry | null | In quadrilateral $K W A D$, the midpoints of $K W$ and $A D$ are $M$ and $N$ respectively. If $M N=\frac{1}{2}(A W+D K)$, prove that $WA$ is parallel to $K D$.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2563 | Number Theory | null | Consider the first $2 n$ natural numbers. Pair off the numbers, as shown, and multiply the two members of each pair. Prove that there is no value of $n$ for which two of the $n$ products are equal.
| null | true | null | null | null | TP_MM_maths_en_COMP | |
2803 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2804 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2805 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2807 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2809 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2811 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2812 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2813 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2814 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2815 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2816 | Combinatorics | null | This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2873 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2874 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2877 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2878 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2881 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2883 | Combinatorics | null | An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2928 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2934 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2935 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2936 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2937 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2938 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2939 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2941 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2942 | Geometry | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2943 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2944 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2945 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
2946 | Combinatorics | null | A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circ... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3052 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3054 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3055 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3056 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3059 | Combinatorics | null | The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it... | null | true | null | null | null | TP_MM_maths_en_COMP | |
3072 | Combinatorics | null | Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
$$
\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text {... | null | true | null | null | null | TP_MM_maths_en_COMP | |
938 | Mechanics | 4. A complex dance
In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strangely defined to obe... | (a) The usual form of Newton's second law $(\vec{F}=m \vec{a})$ breaks down when we go into a rotating frame, where both the centrifugal and Coriolis forces become important to account for. Newton's second law then takes the form
$$
\vec{F}=m(\vec{a}+2 \vec{v} \times \vec{\Omega}+\vec{\Omega} \times(\vec{\Omega} \time... | null | false | null | null | null | TP_TO_physics_en_COMP | |
943 | Mechanics | 4. A complex dance
In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strangely defined to obe... | (f) If the energy of a wave is $E=\hbar \omega$ and the momentum is $p=\hbar k$, show that the dispersion relation found in part (e) resembles the classical expectation for the kinetic energy of a particle, $\mathrm{E}=\mathrm{mv}^{2} / \mathbf{2}$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
946 | Electromagnetism | 5. Polarization and Oscillation
In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations.
Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster than the classic... | (b) Laplace's equation is a second order differential equation
$$
\nabla^{2} \phi=\frac{\partial^{2} \phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}}+\frac{\partial^{2} \phi}{\partial z^{2}}=0
\tag{8}
$$
Solutions to this equation are called harmonic functions. One of the most important properties satis... | null | false | null | null | null | TP_TO_physics_en_COMP | |
963 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (a) The homogeneity of space and time imply that the laws of physics are the same no matter where in space and time you are. In other words, they do not depend on a choice of origin for coordinates $x$ and $t$. Use this fact to show that $\frac{\partial X}{\partial x}$ is independent of the position $x$ and $\frac{\par... | null | false | null | null | null | TP_TO_physics_en_COMP | |
964 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (b) The isotropy of space implies that there is no preferred direction in the universe, i.e., that the laws of physics are the same in all directions. Use this to study the general coordinate transformations $X, T$ after setting $x \rightarrow-x$ and $x^{\prime} \rightarrow-x^{\prime}$ and conclude that $A(v), D(v)$ ar... | null | false | null | null | null | TP_TO_physics_en_COMP | |
965 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (c) The principle of relativity implies that the laws of physics are agreed upon by observers in inertial frames. This implies that the general coordinate transformations $X, T$ are invertible and their inverses have the same functional form as $X, T$ after setting $v \rightarrow-v$. Use this fact to show the following... | null | false | null | null | null | TP_TO_physics_en_COMP | |
966 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (d) Use the previous results and the fact that the location of the $F^{\prime}$ frame may be given by $x=v t$ in the $F$ frame to conclude that the coordinate transformations have the following form:
$$
\begin{aligned}
x^{\prime} & =A(v) x-v A(v) t \\
t^{\prime} & =-\left(\frac{A(v)^{2}-1}{v A(v)}\right) x+A(v) t
\end... | null | false | null | null | null | TP_TO_physics_en_COMP | |
967 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (e) Assume that a composition of boosts results in a boost of the same functional form. Use this fact and all the previous results you have derived about these generalized boosts to conclude that
$$
\frac{A(v)^{2}-1}{v^{2} A(v)}=\kappa .
$$
for an arbitrary constant $\kappa$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
968 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (f) (1 point) Show that $\kappa$ has dimensions of (velocity $)^{-2}$, and show that the generalized boost now has the form
$$
\begin{aligned}
x^{\prime} & =\frac{1}{\sqrt{1-\kappa v^{2}}}(x-v t) \\
t^{\prime} & =\frac{1}{\sqrt{1-\kappa v^{2}}}(t-\kappa v x)
\end{aligned}
$$ | null | false | null | null | null | TP_TO_physics_en_COMP | |
969 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (g) Assume that $v$ may be infinite. Argue that $\kappa=0$ and show that you recover the Galilean boost. Under this assumption, explain using a Galilean boost why this implies that a particle may travel arbitrarily fast. | null | false | null | null | null | TP_TO_physics_en_COMP | |
970 | Modern Physics | 4. Lorentz Boost
In Newtonian kinematics, inertial frames moving relatively to each other are related by the following transformations called Galilean boosts:
$$
\begin{aligned}
x^{\prime} & =x-v t \\
t^{\prime} & =t
\end{aligned}
$$
In relativistic kinematics, inertial frames are similarly related by the Lorentz bo... | (h) Assume that $v$ must be smaller than a finite value. Show that $1 / \sqrt{\kappa}$ is the maximum allowable speed, and that this speed is frame invariant, i.e., $\frac{d x^{\prime}}{d t^{\prime}}=\frac{d x}{d t}$ for something moving at speed $1 / \sqrt{\kappa}$. Experiment has shown that this speed is $c$, the spe... | null | false | null | null | null | TP_TO_physics_en_COMP | |
975 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (a) The electromagnetic modes travel through the ends, $x=0$ and $x=L$, of the resistor. Show that the wavevectors corresponding to periodic waves on the interval $[0, L]$ are $k_{n}=\frac{2 \pi n}{L}$.
Then, show that the number of states per angular frequency is $\frac{d n}{d \omega_{n}}=\frac{L}{2 \pi c^{\prime}}$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
976 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (b) Each mode $n$ in the resistor can be thought of as a species of particle, called a bosonic collective mode. This particle obeys Bose-Einstein statistics: the average number of particles $\left\langle N_{n}\right\rangle$ in the mode $n$ is
$$
\left\langle N_{n}\right\rangle=\frac{1}{\exp \frac{\hbar \omega_{n}}{k T... | null | false | null | null | null | TP_TO_physics_en_COMP | |
977 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (c) By analogy to the photon, explain why the energy of each particle in the mode $n$ is $\hbar \omega_{n}$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
978 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (d) Using parts (a), (b), and (c), show that the average power delivered to the resistor (or produced by the resistor) per frequency interval is
$$
P[f, f+d f] \approx k T d f .
\tag{6}
$$
Here, $f=\omega / 2 \pi$ is the frequency. $P[f, f+d f]$ is known as the available noise power of the resistor. (Hint: Power is d... | null | false | null | null | null | TP_TO_physics_en_COMP | |
979 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (a) Assume that resistors $R$ and $r$ are in series with a voltage $V . R$ and $V$ are fixed, but $r$ can vary. Show the maximum power dissipation across $r$ is
$$
P_{\max }=\frac{V^{2}}{4 R} .
\tag{7}
$$
Give the optimal value of $r$ in terms of $R$ and $V$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
980 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (b) If the average power per frequency interval delivered to the resistor $r$ is $\frac{d\left\langle P_{\max }\right\rangle}{d f}=$ $\frac{d E}{d f}=k T$, show that $\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R$. | null | false | null | null | null | TP_TO_physics_en_COMP | |
981 | Electromagnetism | 2. Johnson-Nyquist noise
In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency, $f$ ) density of noise produced by a resistor, $R$ :
$$
\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R
\tag{2}
$$
Here, \langle\rangle denotes an average ove... | (a) Explain why no Johnson-Nyquist noise is produced by ideal inductors or capacitors. There are multiple explanations; any explanation will be accepted. (Hint: the impedance of an ideal inductor or capacitor is purely imaginary.) | null | false | null | null | null | TP_TO_physics_en_COMP | |
1575 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 4. Show that the proper acceleration of the particle, $a^{\prime} \equiv g=F / m$, is a constant. The proper acceleration is the acceleration of the particle measured in the instantaneous proper frame. | null | false | null | null | null | TP_TO_physics_en_COMP | |
1578 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 1. At a certain moment, the time experienced by the particle is $\tau$. What reading $t_{0}$ on a stationary clock located at $x=0$ will be observed by the particle? After a long period of time, does the observed reading $t_{0}$ approach a certain value? If so, what is the value? | null | false | null | null | null | TP_TO_physics_en_COMP | |
1579 | Modern Physics | Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a... | 2. Now consider the opposite point of view. If an observer at the initial point $(x=0)$ is observing the particle's clock when the observer's time is $t$, what is the reading of the particle's clock $\tau_{0}$ ? After a long period of time, will this reading approach a certain value? If so, what is the value? | null | false | null | null | null | TP_TO_physics_en_COMP | |
1591 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 1. Prove that the magnitude of magnetic moment $\mu$ is always constant under the influence of a magnetic field $\boldsymbol{B}$. For a special case of stationary (constant) magnetic field, also show that the angle between $\boldsymbol{\mu}$ and $\boldsymbol{B}$ is constant.
(Hint: You can use properties of vector pr... | null | false | null | null | null | TP_TO_physics_en_COMP | |
1593 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 1. Show that the time evolution of the magnetic moment follows the equation
$$
\left(\frac{d \boldsymbol{\mu}}{d t}\right)_{r o t}=-\gamma \boldsymbol{\mu} \times \boldsymbol{B}_{e f f}
$$
where $\boldsymbol{B}_{\text {eff }}=\boldsymbol{B}-\frac{\omega}{\gamma} \boldsymbol{k}^{\prime}$ is the effective magnetic fie... | null | false | null | null | null | TP_TO_physics_en_COMP | |
1595 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 3. Now, let us consider the case of a time-varying magnetic field. Besides a constant magnetic field, we also apply a rotating magnetic field $\boldsymbol{b}(t)=b(\cos \omega t \boldsymbol{i}+\sin \omega t \boldsymbol{j})$, so $\boldsymbol{B}=B_{0} \boldsymbol{k}+\boldsymbol{b}(t)$. Show that the new Larmor precession... | null | false | null | null | null | TP_TO_physics_en_COMP | |
1598 | Modern Physics | All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual classical formali... | 2. Determine the angle $\alpha$ that $\boldsymbol{\mu}$ makes with $\boldsymbol{B}_{\text {eff }}$. Also, prove that the magnetization varies with time as
$$
M(t)=N \mu(\cos \Omega t) .
$$ | null | false | null | null | null | TP_TO_physics_en_COMP | |
950 | Electromagnetism | 5. Polarization and Oscillation
In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations.
Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster than the classic... | (f) Now suppose that we attach the point-like charge to a wall with a rod of length a. Any perturbation from the equilibrium will cause a perturbation of the polarization of the sphere. Prove that this equilibrium is stable and find the frequency of oscillation around it. The charge and rod have masses $m$ and $M$, res... | null | false | null | null | null | TP_MM_physics_en_COMP | |
992 | Mechanics | 4. The fundamental rocket equation
In this problem, we will investigate the acceleration of rockets.
Rocket repulsion
In empty space, an accelerating rocket must "throw" something backward to gain speed from repulsion. Assume there is zero gravity.
The rocket ejects fuel from its tail to propel itself forward. From... | (c) Discuss the factors that limit the final speed of the rocket. | null | false | null | null | null | TP_MM_physics_en_COMP | |
999 | Modern Physics | X-ray Diffraction from a crystal.
We wish to study X-ray diffraction by a cubic crystal lattice. To do this we start with the diffraction of a plane, monochromatic wave that falls perpendicularly on a 2-dimensional grid that consists of $\mathrm{N}_{1} \times \mathrm{N}_{2}$ slits with separations $\mathrm{d}_{1}$ and... | c - Show that this so-called Bragg reflection yields the same conditions for the maxima as those that you found in $b$. | null | false | null | null | null | TP_MM_physics_en_COMP | |
1002 | Electromagnetism | Electric experiments in the magnetosphere of the earth.
In May 1991 the spaceship Atlantis will be placed in orbit around the earth. We shall assume that this orbit will be circular and that it lies in the earth's equatorial plane.
At some predetermined moment the spaceship will release a satellite $S$, which is att... | $a_{2}$ - Discuss the stability of the equilibrium for each case. | null | false | null | null | null | TP_MM_physics_en_COMP | |
1003 | Electromagnetism | Electric experiments in the magnetosphere of the earth.
In May 1991 the spaceship Atlantis will be placed in orbit around the earth. We shall assume that this orbit will be circular and that it lies in the earth's equatorial plane.
At some predetermined moment the spaceship will release a satellite $S$, which is att... | $\mathrm{b}$ - Express the period of the swinging in terms of the period of revolution of the system around the earth. | null | false | null | null | null | TP_MM_physics_en_COMP | |
1025 | Optics | a) Consider a plane-parallel transparent plate, where the refractive index, $n$, varies with distance, $z$, from the lower surface (see figure). Show that $n_{A} \sin \alpha=n_{B} \sin \beta$. The notation is that of the figure.
| null | false | null | null | null | TP_MM_physics_en_COMP | ||
1026 | Optics |
Context question:
a) Consider a plane-parallel transparent plate, where the refractive index, $n$, varies with distance, $z$, from the lower surface (see figure). Show that $n_{A} \sin \alpha=n_{B} \sin \beta$. The notation is that of the figure.
<img_4366>
Context answer:
\boxed{证明题}
| b) Assume that you are standing in a large flat desert. At some distance you see what appears to be a water surface. When you approach the "water" is seems to move away such that the distance to the "water" is always constant. Explain the phenomenon. | null | false | null | null | null | TP_MM_physics_en_COMP | |
1045 | Electromagnetism | Electrostatic lens
Consider a uniformly charged metallic ring of radius $R$ and total charge $q$. The ring is a hollow toroid of thickness $2 a \ll R$. This thickness can be neglected in parts A, B, C, and E. The $x y$ plane coincides with the plane of the ring, while the $z$-axis is perpendicular to it, as shown in F... | C.3 Is the equation of a thin optical lens
$$
\frac{1}{b}+\frac{1}{c}=\frac{1}{f}
$$
fulfilled for the electrostatic lens? Show it by explicitly calculating $1 / b+1 / c$. | null | false | null | null | null | TP_MM_physics_en_COMP | |
1057 | Modern Physics | Particles and Waves
Wave-particle duality, which states that each particle can be described as a wave and vice versa, is one of the central concepts of quantum mechanics. In this problem, we will rely on this notion and just a few other basic assumptions to explore a selection of quantum phenomena covering the two dis... | D.2 The resulting potential energy has a sixfold rotational symmetry axis, i.e., the potential distribution is invariant with respect to a rotation by a multiple of $60^{\circ}$ around the origin. Provide a simple argument to prove that this is indeed the case. | null | false | null | null | null | TP_MM_physics_en_COMP | |
1156 | Optics | Figure 1.1
<img_4357>
A plane monochromatic light wave, wavelength $\lambda$ and frequency $f$, is incident normally on two identical narrow slits, separated by a distance $d$, as indicated in Figure 1.1. The light wave emerging at each slit is given, at a distance $x$ in a direction $\theta$ at time $t$, by
$$
y=a ... | (i) Show that the two waves observed at an angle $\theta$ to a normal to the slits, have a resultant amplitude A which can be obtained by adding two vectors, each having magnitude $a$, and each with an associated direction determined by the phase of the light wave.
Verify geometrically, from the vector diagram, that
... | null | false | null | null | null | TP_MM_physics_en_COMP | |
1157 | Optics | Figure 1.1
<img_4357>
A plane monochromatic light wave, wavelength $\lambda$ and frequency $f$, is incident normally on two identical narrow slits, separated by a distance $d$, as indicated in Figure 1.1. The light wave emerging at each slit is given, at a distance $x$ in a direction $\theta$ at time $t$, by
$$
y=a ... | (ii) The double slit is replaced by a diffraction grating with $N$ equally spaced slits, adjacent slits being separated by a distance $d$. Use the vector method of adding amplitudes to show that the vector amplitudes, each of magnitude $a$, form a part of a regular polygon with vertices on a circle of radius $R$ given ... | null | false | null | null | null | TP_MM_physics_en_COMP | |
1159 | Optics | Figure 1.1
<img_4357>
A plane monochromatic light wave, wavelength $\lambda$ and frequency $f$, is incident normally on two identical narrow slits, separated by a distance $d$, as indicated in Figure 1.1. The light wave emerging at each slit is given, at a distance $x$ in a direction $\theta$ at time $t$, by
$$
y=a ... | (v) Show that the number of principal maxima cannot exceed
$$
\left(\frac{2 d}{\lambda}+1\right)
$$ | null | false | null | null | null | TP_MM_physics_en_COMP | |
1162 | Mechanics | Three particles, each of mass $m$, are in equilibrium and joined by unstretched massless springs, each with Hooke's Law spring constant $k$. They are constrained to move in a circular path as indicated in Figure 3.1.
Figure 3.1
<img_4463>
Context question:
(i) If each mass is displaced from equilibrium by small displ... | (ii) Verify that the system has simple harmonic solutions of the form
$$
u_{n}=a_{n} \cos \omega t
$$
with accelerations, $\left(-\omega^{2} u_{n}\right)$ where $a_{n}(n=1,2,3)$ are constant amplitudes, and $\omega$, the angular frequency, can have 3 possible values,
$$
\omega_{o} \sqrt{3}, \omega_{o} \sqrt{3} \text... | null | false | null | null | null | TP_MM_physics_en_COMP |
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