prompt stringlengths 54 14k | response stringlengths 3 6 |
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Problem: ## Task B-1.3.
A ship traveling along a river has covered $24 \mathrm{~km}$ upstream and $28 \mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \mathrm{~km}$ upstream and $21 \mathrm{~km}$ downstream, or half an hour more than for traveling $15 \mathrm{~km}$ upstream ... | 2752. |
Problem: 3. (6 points) A construction company was building a tunnel. When $\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \%$ and reduced the working hours to $80 \%$ of the original. As a result, it took a total of 185 day... | 2316. |
Problem: Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?
Answer: 1. **Restate the problem in a more manageable form:**... | 15848. |
Problem: 4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).
(A) $1<x<7$.
(B) $5 \ll x<7$.
(C) $1<x<\sqrt{7}$.
(D) $5<x<7$ or $1<x<\sqrt{7}$.
Answer: $D$ | 1937. |
Problem: 1. Solve the equation: $\frac{8 x+13}{3}=\frac{19-12 x}{2}-\left(16 x-\frac{7-4 x}{6}\right)$.
Answer: 1. $\frac{8 x+13}{3}=\frac{19-12 x}{2}-\left(16 x-\frac{7-4 x}{6}\right)$
$\frac{8 x+13}{3}=\frac{19-12 x}{2}-16 x+\frac{7-4 x}{6} / \cdot 6$
1 BOD
$2 \cdot(8 x+13)=3 \cdot(19-12 x)-96 x+7-4 x \quad 1$ BOD... | 1490. |
Problem: A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?
Answer: Let the lengths of the legs be $a$ and $b$, and the length of the hypotenuse be $c$. According to the Pythagorean theorem, we then have
... | 10111. |
Problem: Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$
Answer: 1. Define the integral \( I_n(x) = \int_x^{2x} e^{-t^n} \, dt \). We need to find the value of \( x \) that maximizes \( I_n(x) \).
2. To find the maximum, we first compute the... | 3596. |
Problem: 11. Given that the internal angles $A, B, C$ of $\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\sqrt{3} b \cos \frac{A+B}{2}=c \sin B$.
(1) Find the size of $\angle C$;
(2) If $a+b=\sqrt{3} c$, find $\sin A$.
Answer: (1) $\sqrt{3} \sin B \cos \frac{A+B}{2}=\sin C \sin B \Rightarrow \sqrt{3} \... | 12314. |
Problem: Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\mathbb{C}$.
Answer: ## Solution.
If between every two digits of the number 14641 we write 2013 zeros, we get a number of the form $1 \un... | 3152. |
Problem: 6. As shown in Figure 2, let $P$ be a point inside the equilateral $\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $B... | 6263. |
Problem: 8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation
$$
f(x)+f(y)=f(x+y)-x y-1 .
$$
If $f(1)=1$, then the number of integers $n$ (where $n \neq 1$) that satisfy $f(n)=n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) infinitely many.
(30th American High School Mathematics Examinat... | 4439. |
Problem: 9.6. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{36-4 \sqrt{5}} \sin x+2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer.
Answer: Answer: -27 . Instructions. Exact answer: $4 \sqrt{5}-36$. | 8192. |
Problem: Given $0 \leqslant x, y, z \leqslant 1$, solve the equation:
$$\frac{x}{1+y+z x}+\frac{y}{1+z+x y}+\frac{z}{1+x+y z}=\frac{3}{x+y+z} .$$
Answer: 3. It is not difficult to prove: $\frac{x}{1+y+z x} \leqslant \frac{1}{x+y+z}$, $\frac{y}{1+z+x y} \leqslant \frac{1}{x+y+z}$, $\frac{z}{1+x+y z} \leqslant \frac{1}{x... | 7436. |
Problem: ## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}
$$
Answer: ## Solution
$$
\int_{0}^{3 / 2} \frac{x^{2} \cdot d x}{\sqrt{9-x^{2}}}=
$$
Substitution:
$$
\begin{aligned}
& x=3 \sin t \Rightarrow d x=3 \cos t d t \\
& x=0 \Rightarrow t=\arcsin \... | 3209. |
Problem: Example 6 The rules of a "level-up game" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:
(1) What is the maximum number of levels a person can pass in this game?
(2) What i... | 5569. |
Problem: 2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.
Answer: 2. This problem is equivalent to finding the smallest positive integer $b$, such that the equati... | 3821. |
Problem: Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the ... | 7075. |
Problem: 7.1. Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \cdot 2 \cdot \ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; ... | 5080. |
Problem: Example 1 (Question from the 13th "Hope Cup" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\sqrt[5]{x}+1=0$ are $\alpha, \beta$ respectively, then $\alpha+\beta$ equals ( ).
A. -1
B. $-\frac{1}{2}$
C. $\frac{1}{2}$
D. 1
Answer: Solution: Choose A. Reason: Consider the function ... | 3146. |
Problem: Example 5 Given that $x_{1}, x_{2}, \cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{10}^{2}$.
Answer: Solve: The number of ways to write 2005 as the sum of 10 positive integers is finite, so there must be a w... | 2177. |
Problem: Four, (50 points) In an $n \times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.
---
The translation preserves the original text's format... | 8300. |
Problem: 1. If the set
$$
A=\{1,2, \cdots, 10\}, B=\{1,2,3,4\},
$$
$C$ is a subset of $A$, and $C \cap B \neq \varnothing$, then the number of such subsets $C$ is $(\quad)$.
(A) 256
(B) 959
(C) 960
(D) 961
Answer: - 1. C.
From the fact that there are $2^{6}$ subsets $C$ satisfying $C \cap B=\varnothing$, we know that ... | 2080. |
Problem: Augusto has a wire that is $10 \mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:
. MATHEMATICE YoUTH CLUE A. 25
B. 31
C. 37
D. 43
Answer: Reference answer: B | 2843. |
Problem: Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?
Answer: Solution. Since 16 teachers do not drink either coffee or tea, we get that $50-16=34$ teachers drink either... | 1156. |
Problem: 12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.
Answer: ## Solution.
Let $a$ be the given side of the triangle, $a=15$, $b$ and $c$ be the other two sides, $b+... | 2203. |
Problem: 3. Let $AB$ be a chord of the unit circle $\odot O$. If the area of $\odot O$ is exactly equal to the area of the square with side $AB$, then $\angle AOB=$ $\qquad$ (to 0.001 degree).
Answer: 3. $124.806^{\circ}$ | 9811. |
Problem: 83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\qquad$. $(a+b$ and $b+a$ are considered the same equation)
Answer: Answer: 1536 | 12876. |
Problem: On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.
An... | 4301. |
Problem: 4. The number of real solutions to the equation $\left|x^{2}-3 x+2\right|+\left|x^{2}+2 x-3\right|=11$ is ( ).
(A) 0
(B) 1
(C) 2
(D) 4
Answer: 4. C.
The original equation is $|x-1|(|x-2|+|x+3|)=11$.
By discussing four cases: $x \leqslant -3$, $-3 < x \leqslant 2$, $2 < x \leqslant 1$, and $x > 1$, we know tha... | 7448. |
Problem: ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$
Answer: ## Solution
$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}=\lim _{n \rightarrow \infty} \frac{\left((n... | 3516. |
Problem: Condition of the problem
Find the derivative.
$$
y=\frac{1}{24}\left(x^{2}+8\right) \sqrt{x^{2}-4}+\frac{x^{2}}{16} \arcsin \frac{2}{x}, x>0
$$
Answer: ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{1}{24}\left(x^{2}+8\right) \sqrt{x^{2}-4}+\frac{x^{2}}{16} \arcsin \frac{2}{x}\right)^{\prime}= \\
&... | 6223. |
Problem: \section*{Problem 5 - 071225}
All ordered pairs of real numbers \((x, y)\) are to be determined for which the system of equations
\[
\begin{aligned}
x \cdot\left(a x^{2}+b y^{2}-a\right) & =0 \\
y \cdot\left(a x^{2}+b y^{2}-b\right) & =0
\end{aligned}
\]
is satisfied. Here, \(a\) and \(b\) are real numbers ... | 4072. |
Problem: 6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\f... | 4126. |
Problem: 40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers.
Answer: 40. The sum of three consecutive odd numbers is an odd number, and if the fourth power is an odd number, then the base is also odd. Therefore, single-digit numbers shoul... | 2434. |
Problem: Two is $10 \%$ of $x$ and $20 \%$ of $y$. What is $x - y$?
$(\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D}) \ 10 \qquad (\mathrm {E})\ 20$
Answer: $2 = \frac {1}{10}x \Longrightarrow x = 20,\quad 2 = \frac{1}{5}y \Longrightarrow y = 10,\quad x-y = 20 - 10=10 \mathrm{(D)... | 1036. |
Problem: 19. Given $m \in\{11,13,15,17,19\}$, $n \in\{1999,2000, \cdots, 2018\}$.
Then the probability that the unit digit of $m^{n}$ is 1 is ( ).
(A) $\frac{1}{5}$
(B) $\frac{1}{4}$
(C) $\frac{3}{10}$
(D) $\frac{7}{20}$
(E) $\frac{2}{5}$
Answer: 19. E.
Consider different cases.
When $m=11$, $n \in\{1999,2000, \cdots,... | 3280. |
Problem: 1. (6 points) Today is January 31, 2015, welcome to the 2015 "Spring Cup" Final. The calculation result of the expression $\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}$ is
Answer: 【Solution】Solve: $\frac{\frac{2015}{1}+\frac{2015}{0.31}}{1+0.31}$
$$
\begin{array}{l}
=\frac{\left(\frac{2015}{1}+\frac{2015}{0... | 1761. |
Problem: Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?
Answer: 1. **Assume the side length of the regular hexagon is 1.**
- Let the side length of the hexagon be \( s = 1 \).
2. **Calculate the area of the regular hexagon.**
- The formu... | 2413. |
Problem: Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
Answer: 1. **Define the function \( f(P) \):**
For any... | 5004. |
Problem: 1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is
Answer: $-1 . \frac{\sqrt{2}}{2}$.
From the problem, we know that the tetrahedron $ABCD$ is a regular tetrahedron. Therefore, finding... | 5097. |
Problem: The function $f$ maps the set of positive integers into itself, and satisfies the equation
$$
f(f(n))+f(n)=2 n+6
$$
What could this function be?
Answer: I. solution. The function $f(n)=n+2$ is suitable because its values are positive integers, and for any positive integer $n$,
$$
f(f(n))+f(n)=((n+2)+2)+(n+2... | 3680. |
Problem: 30. Find the remainder when the 2018-digit number $\underbrace{\overline{55 \cdots}}_{2018 \text { 555 }}$ is divided by 13.
Answer: Reference answer: 3 | 2367. |
Problem: 1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).
A. 1
B. 2
C. 3
D. 4
Answer: 一、1. B. The intersection point refers to the solution of the system of equations
$$
\left\{\begin{array}{l}
a x+b y+c=0 \\
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
\end{array}\right.
$$... | 2453. |
Problem: 1. Let the universal set be the set of real numbers. If $A=\{x \mid \sqrt{x-2} \leqslant 0\}, B=\left\{x \mid 10^{x^{2}-2}=10^{x}\right\}$, then $A \cap \bar{B}$ is
A. $\{2\}$
B. $\{-1\}$
C. $\{x \mid x \leqslant 2\}$
D. $\varnothing$
Answer: $$
A=\{2\}, x^{2}-2=x \Rightarrow x^{2}-x-2=(x+1)(x-2)=0 \Rightarrow... | 1597. |
Problem: 4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \perp A M$, then $\angle B A C$ equals ( ).
(A) $30^{\circ}$
(B) $45^{\circ}$
... | 3545. |
Problem: 7.242. $\left(16 \cdot 5^{2 x-1}-2 \cdot 5^{x-1}-0.048\right) \lg \left(x^{3}+2 x+1\right)=0$.
Answer: Solution.
Domain: $x^{3}+2 x+1>0$.
From the condition $16 \cdot 5^{2 x-1}-2^{x-1}-0.048=0$ or $\lg \left(x^{3}+2 x+1\right)=0$. Rewrite the first equation as
$\frac{16}{5} \cdot 5^{2 x}-\frac{2}{5} \cdot 5... | 3122. |
Problem: Example 1 In $\triangle ABC$, it is known that $x \sin A + y \sin B + z \sin C = 0$. Find the value of $(y + z \cos A)(z + x \cos B)(x + y \cos C) + (y \cos A + z)(z \cos B + x)(x \cos C + y)$.
Answer: In $\triangle A B C$, $\sin C=\sin (A+B)=\sin A \cos B+\cos A \sin B$. Substituting the known conditions, we ... | 10098. |
Problem: In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$
Ans... | 1494. |
Problem: 11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base.
Answer: Solution.
For a regular pyramid $AB=BC=AC=a \quad$ (Fig. 11.7). $\angle ASC=90^{\circ}, SA=SC=SB$. This means that $\angle SAC=45^{\... | 5905. |
Problem: The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?
... | 1443. |
Problem: A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It t... | 2918. |
Problem: Example 11 Let $x>0, y>0, \sqrt{x}(\sqrt{x}+2 \sqrt{y})$ $=\sqrt{y}(6 \sqrt{x}+5 \sqrt{y})$. Find the value of $\frac{x+\sqrt{x y}-y}{2 x+\sqrt{x y}+3 y}$.
Answer: Solution: From the given, we have
$$
(\sqrt{x})^{2}-4 \sqrt{x} \sqrt{y}-5(\sqrt{y})^{2}=0,
$$
which is $(\sqrt{x}-5 \sqrt{y})(\sqrt{x}+\sqrt{y})=0... | 1878. |
Problem: 9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?
Answer: Solution: $1 / 2$
Imagine placing the tetrahedron $A B C D$ flat on a table with vertex $A$ at the top. By vectors or otherwis... | 12311. |
Problem: Let's determine all the triples of numbers $(x, y, m)$ for which
$$
-2 x + 3 y = 2 m, \quad x - 5 y = -11
$$
and $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number.
Answer: From the second equation, $x=5 y-11$. Since $x$ is negative according to our condition, the equality can onl... | 1668. |
Problem: ## Zadatak B-1.2.
Na slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.
Answer: ## Rješenje.
Zbroj mjera unutrašnjih kutova pravi... | 8705. |
Problem: The equation of line $\ell_1$ is $24x-7y = 319$ and the equation of line $\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\ell_1$ and $\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Deter... | 4287. |
Problem: Let's determine $m$ such that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
---
Determine $m$ so that the expression
$$
m x^{2}+(m-1) x+m-1
$$
is negative for all values of $x$.
Answer: For the given expression to be negative for all values of $x$, it is necessary that the c... | 3490. |
Problem: 2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.
Answer: 2. Solution. Since the desired integer solution $x$ is divisible by 10 and 12, it is divisible by 60, hence it can be written in the f... | 8527. |
Problem: Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \leqslant x \leqslant 0)$ is $(\quad)$.
(A) $(-\infty, 4]$
(B) $[3,12]$
(C) $[-12,4]$
(D) $[4,12]$
Answer: Solve $y=-(x+1)^{2}+4$, and $-5 \leqslant x \leqslant 0$, so, the range of the function $y=-x^{2}-2 x+3(-5 \leqslant$ $x \leqslant 0)$ is $[-12,4]$, ... | 1244. |
Problem: 4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,
$$\begin{array}{l}
\vec{f}_{1}=\overrightarrow{O A}=\sqrt{2}\left(\cos 45^{\circ}+i \sin 45^{\circ}\right) \\
\vec{f}_{2}=\overrightarrow{O B}=2\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\rig... | 6611. |
Problem: 6. Given that $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. If $t \in[0,1]$, then
$$
|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|
$$
the minimum value is ( ).
(A) $2 \sqrt{19... | 10436. |
Problem: All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(... | 1976. |
Problem: 1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$.
Answer: Solution. By adding the three equations, we obtain
$$
a^{2}+b^{2}+c^{2}+a+b+c=a^{2}+b^{2}+c^{2}, \text { i.e., } a+b+c=0
... | 8464. |
Problem: 2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).
(A) $\frac{\sqrt{5}+1}{2}$
(B) $\frac{\sqrt{5}-1}{2}$
(C) 1
(D) 2
Answer: 2. A. Since $\triangle B O C \sim \triangle A B C$, we have $\frac{B O}{... | 3581. |
Problem: V-2 If one side of the rectangle is reduced by $3 \mathrm{~cm}$, and the other side is reduced by $2 \mathrm{~cm}$, we get a square whose area is $21 \mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.
=b-\log _{a} a-\log _... | 3679. |
Problem: 4. Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$.
Answer: 4. First, prove a lemm... | 5496. |
Problem: In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?
Answer: Solution. The first distributor can choose 5 out of 10 mailboxes in $... | 4360. |
Problem: 18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his origi... | 2987. |
Problem: 53. How many four-digit numbers contain at least one even digit?
Answer: 53. From four-digit numbers, we need to discard all those numbers that do not have a single even digit. We will get: $9 \cdot 10 \cdot 10 \cdot 10-5 \cdot 5 \cdot 5 \cdot 5 \cdot 5=8375$ numbers. | 3067. |
Problem: 1. The range of the function $f(x)=\sin x+\cos x+\tan x+$ $\arcsin x+\arccos x+\arctan x$ is $\qquad$ .
Answer: $=、 1 \cdot\left[-\sin 1+\cos 1-\tan 1+\frac{\pi}{4}, \sin 1+\cos 1+\tan 1+\frac{3 \pi}{4}\right]$.
Obviously, $f(x)=\sin x+\cos x+\tan x+\arctan x+\frac{\pi}{2}$, $x \in[-1,1]$.
Below we prove: $g(... | 8193. |
Problem: Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
[i]Romania[/i]
Answer: To determine the largest possible value of the sum
\[ S(n) = |a_2 - a_1| + |a_3 - a_2| + \... | 15154. |
Problem: There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$... | 16385. |
Problem: 10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid.
Answer: Solution.
Let in trapezoid $A B C D$ (Fig. 10.108) $A B=C D, A C \perp B D, O$ - the intersection point of $A C$ and $B D, C K$ - the height of the trapezoid... | 4659. |
Problem: 8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
Answer: 8. Solution: $\because 7 \equiv-1(\bmod 4), \therefore 7^{7^{7}} \equiv$ $-1(\bmod 4)(98$ sevens$)$, let... | 10251. |
Problem: 1. Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$. (Here, a seque... | 4981. |
Problem: Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \... | 16385. |
Problem: 9.171. $0.6^{\lg ^{2}(-x)+3} \leq\left(\frac{5}{3}\right)^{2 \lg x^{2}}$.
Answer: ## Solution.
Domain of definition: $x<0$.
Since $\lg x^{2 k}=2 k \lg |x|$, taking into account the domain of definition, we can rewrite the given inequality as
$\left(\frac{3}{5}\right)^{\lg ^{2}(-x)+3} \leq\left(\frac{3}{5}\r... | 5262. |
Problem: 2. Find all integer solutions of the inequality
$$
x^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \leq 0
$$
Answer: # Solution:
$\left(x^{2} y^{2}+2 x y z+z^{2}\right)+\left(y^{2} z^{2}+2 x y z+x^{2}\right)-38(x y+z)-40(y z+x)+761 \leq 0$.
$(x y+z)^{2}+(y z+x)^{2}-38(x y+z)-40(y z+x)+76... | 12172. |
Problem: Solve the triangle whose area $t=357.18 \mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.
Answer: If $2 s=a+b+c$, then the area of the triangle is:
$$
t=\sqrt{s \cdot(s-a) \cdot(s-b) \cdot(s-c)}
$$
Since
$$
a: b: c=4: 5: 6
$$
therefore
$$
b=\frac{5 a}{4}, c=\frac{6 a}{4} \text { and } ... | 4484. |
Problem: Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \]
Answer: We are given the equation:
\[ 2^m p^2 + 1 = q^5 \]
where \( m \) is a positive integer and \( p \) and \( q \) are primes. We need to find all possible triples \((m, p, q)\).
1. **Rewrite the... | 7635. |
Problem: 15. As shown in the figure, the area of square $\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter.... | 7875. |
Problem: 25. Anna, Bridgit and Carol run in a $100 \mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?
A $37 \mathrm{~m}$
B $41 \m... | 3577. |
Problem: Given the complex number $z$ has a modulus of 1. Find
$$
u=\frac{(z+4)^{2}-(\bar{z}+4)^{2}}{4 i} \text {. }
$$
the maximum value.
Answer: $$
\begin{array}{l}
\text { Given }|z|=1 \text {, we can set } z=\cos x+i \sin x \text {. Then, } \\
u=(4+\cos x) \sin x \text {. } \\
\text { By } u^{2}=(4+\cos x)^{2} \si... | 8192. |
Problem: ## Problem 1
Perform the calculations:
a) $7 \cdot 147 - 7 \cdot 47$ (1p)
b) $(2+4+6+8+\cdots+50)-(1+3+5+7+\cdots+49)$ (2p)
c) $10 \cdot 9^{2} : 3^{2} - 3^{4} \quad(2 \text{p})$
d) $(\overline{a b} + \overline{b c} + \overline{c a}) : (a + b + c) \quad$ (2p)
Answer: ## Problem 1
a) 7400 .................... | 2126. |
Problem: 1. The curve $(x+2 y+a)\left(x^{2}-y^{2}\right)=0$ represents three straight lines intersecting at one point on the plane if and only if
A. $a=0$
B. $a=1$
C. $a=-1$
D. $a \in \mathbf{R}$
Answer: Obviously, the three lines are $x+2y+a=0$, $x+y=0$, and $x-y=0$. Since $x+y=0$ and $x-y=0$ intersect at point $(0,0)... | 1619. |
Problem: 9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?
Answer: 9.27 The first passenger can exit at any of the 16 stops, as can the second, i.e., for two passengers there are $16^{2}$ possibi... | 2566. |
Problem: 2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).
(A) the I quadrant
(B) the II quadrant
(C) the III quadrant
(D) the IV quadrant
Answer: 2. (C).
$$
\begin{array}{... | 3146. |
Problem: Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
$ \text{(A)}\ 10\qquad\text{(B)}\ 20\qqu... | 1038. |
Problem: Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.
How many games did father play with uncle?
(M. Volfová)
Answer: Father lost four times, so... | 1091. |
Problem: ## Problem 4
Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.
Selected problems by Prof. Cicortas Marius
Note: a) The actual ... | 5478. |
Problem: Problem 5.3. In five of the nine circles in the picture, the numbers 1, 2, 3, 4, 5 are written. Replace the digits $6, 7, 8, 9$ in the remaining circles $A, B, C, D$ so that the sums of the four numbers along each of the three sides of the triangle are the same.
=2f(x)$ for $\forall$ $x\in \mathbb{R}$ and $f(x)=x(x-1)$ for $\forall$ $x\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\geq -\frac{8}{9}$ is true for
$\forall$ $x\in (-\infty , m]$.
Answer: 1. **Given Condit... | 5537. |
Problem: 18.3.19 $\star \star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \leqslant b \leqslant c$.
Answer: Given $44<\sqrt{2005}<45,25<\sqrt{\frac{2005}{3}}<26$, so
$$
26 \leqslant c \leqslant 44 .
$$
$a^{2}+b^{2}=0(\bmod 3)$, if and only if $a=b=0(\bmod 3)$. Therefore, ... | 9916. |
Problem: 5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $30^{\circ}$.
Answer: Answer: $14... | 11684. |
Problem: 4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$.
Answer: Solution. Let $A=k a+r, B=... | 2238. |
Problem: 7. Given that $z$ is a complex number, and $|z|=1$. When $\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\qquad$ or . $\qquad$
Answer: 7. $-\frac{1}{4} \pm \frac{\sqrt{15}}{4}$ i.
Notice,
$$
\begin{array}{l}
\left|1+z+3 z^{2}+z^{3}+z^{4}\right| \\
=\left|\frac{1}{z^{2}}+\... | 4426. |
Problem: Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance betw... | 5386. |
Problem: 5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\sqrt{1+a^{2}}+\sqrt{1+2 b}$ the integer part is
Answer: 5. 2
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 3225. |
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