Unnamed: 0 int64 0 61 | text stringlengths 585 3.31k |
|---|---|
0 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 1. Find all prime numbers $p$ for which there exist positive integers $x, y$ and $z$ such that the number
$$
x^{p}+y^{p}+z^{p}-x-y-z
$$
is a product of exactly three distinct prime numbers.
Assistant:
So... |
1 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 3. Triangle $A B C$ is such that $A B<A C$. The perpendicular bisector of side $B C$ intersects lines $A B$ and $A C$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $A B C$, and ... |
2 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 1. A trapezoid $A B C D(A B \| C D, A B>C D)$ is circumscribed. The incircle of the triangle $A B C$ touches the lines $A B$ and $A C$ at the points $M$ and $N$, respectively. Prove that the incenter of the ... |
3 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Let $a, b$ and $c$ be positive real numbers. Prove that
$$
\frac{8}{(a+b)^{2}+4 a b c}+\frac{8}{(b+c)^{2}+4 a b c}+\frac{8}{(c+a)^{2}+4 a b c}+a^{2}+b^{2}+c^{2} \geq \frac{8}{a+3}+\frac{8}{b+3}+\frac{8}{... |
4 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Let $n$ three-digit numbers satisfy the following properties:
(1) No number contains the digit 0 .
(2) The sum of the digits of each number is 9 .
(3) The units digits of any two numbers are different.... |
5 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 4. Let $A B C$ be an acute triangle, $A^{\prime}, B^{\prime}$ and $C^{\prime}$ be the reflections of the vertices $A, B$ and $C$ with respect to $B C, C A$, and $A B$, respectively, and let the circumcircles... |
6 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: # Problem 1
Let $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that
$$
\frac{a}{b}+\frac{b}{a}+\frac{b}{c}+\frac{c}{b}+\frac{c}{a}+\frac{a}{c}+6 \geq 2 \sqrt{2}\left(\sqrt{\frac{1-a}{a}}+\sqrt{... |
7 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## Problem 2
Let the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, and let $t$ be a common tangent of $k_{1}$ and $k_{2}$, that touches $k_{1}$ and $k_{2}$ at $M$ and $N$, respectively. I... |
8 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## Problem 3
On a board there are $n$ nails each two connected by a string. Each string is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with strings ... |
9 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 3. Let $A B C$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $B C$. The lines $B C$ and $A O$ intersect at $E$. Let $s$ be the line through $E$ ... |
10 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 4. Let $M$ be a subset of the set of 2021 integers $\{1,2,3, \ldots, 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a+b-c|>10$. Determine the largest possible ... |
11 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 1. Find all distinct prime numbers $p, q$ and $r$ such that
$$
3 p^{4}-5 q^{4}-4 r^{2}=26
$$
Assistant:
Solution. First notice that if both primes $q$ and $r$ differ from 3 , then $q^{2} \equiv r^{2} \equ... |
12 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Consider an acute triangle $A B C$ with area S. Let $C D \perp A B \quad(D \in A B)$, $D M \perp A C \quad(M \in A C)$ and $\quad D N \perp B C \quad(N \in B C)$. Denote by $H_{1}$ and $H_{2}$ the orthoce... |
13 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 3. Let $a, b, c$ be positive real numbers such that $a b c=1$. Prove that
$$
\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2} \geq 3(a+b+c+1)
$$
When does equali... |
14 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Let $A B C$ be an acute triangle such that $A H=H D$, where $H$ is the orthocenter of $A B C$ and $D \in B C$ is the foot of the altitude from the vertex $A$. Let $\ell$ denote the line through $H$ which ... |
15 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 4. We call an even positive integer $n$ nice if the set $\{1,2, \ldots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of 3 . For ... |
16 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## Problem 1.
Find all pairs $(a, b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of 5 .
Assistant:
Solution. The condition is symmetric so we can assume that $b \leq a$.
The first case is wh... |
17 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## Problem 3.
Alice and Bob play the following game on a $100 \times 100$ grid, taking turns, with Alice starting first. Initially the grid is empty. At their turn, they choose an integer from 1 to $100^{2}$ that is... |
18 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## Problem 4.
Let $A B C$ be an acute triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to $B C$ and let $M$ be the midpoint of $O D$. The points $O_{b}$ and $O_{c}$ are the circumcenters ... |
19 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Let $\triangle A B C$ be a right-angled triangle with $\angle B A C=90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ on $B C$. Let $Z \neq A$ be a point on the line $A B$ with $A B=B Z$. ... |
20 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Let $x, y, z$ be positive integers such that $x \neq y \neq z \neq x$. Prove that
$$
(x+y+z)(x y+y z+z x-2) \geq 9 x y z
$$
When does the equality hold?
Assistant:
Solution. Since $x, y, z$ are distin... |
21 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 3. Let $A B C$ be an acute triangle such that $A B \neq A C$, with circumcircle $\Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $B C$ and $D$ be a point on $\Gamma$ such that $A D \perp B C$. Let $... |
22 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem 2. Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression
$$
A=\frac{2-a^{3}}{a}+\frac{2-b^{3}}{b}+\frac{2-c^{3}}{c}
$$
(x y+y z+z x-2) \geq 9 x y z
$$
When does the equality hold?
Assistant:
Solution. Since $x, y, z$ are distinct positive integers, the r... |
26 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
C1. Consider a regular $2 n+1$-gon $P$ in the plane, where $n$ is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $S E... |
27 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
C3. We have two piles with 2000 and 2017 coins respectively. Ann and Bob take alternate turns making the following moves: The player whose turn is to move picks a pile with at least two coins, removes from that pile... |
28 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G1. Given a parallelogram $A B C D$. The line perpendicular to $A C$ passing through $C$ and the line perpendicular to $B D$ passing through $A$ intersect at point $P$. The circle centered at point $P$ and radius $P... |
29 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
Problem G4. Let $A B C$ be a scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $B C$ and $D$ be a point on $\Gamma$ such that $A D \perp B C$. Let $T$ be a point such that ... |
30 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
NT1. Determine all sets of six consecutive positive integers such that the product of two of them, added to the the product of other two of them is equal to the product of the remaining two numbers.
Assistant:
Sol... |
31 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
NT3. Find all pairs of positive integers $(x, y)$ such that $2^{x}+3^{y}$ is a perfect square.
Assistant:
Solution. In order for the expression $2^{x}+3^{y}$ to be a perfect square, a positive integer $t$ such tha... |
32 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
NT5. Find all positive integers $n$ such that there exists a prime number $p$, such that
$$
p^{n}-(p-1)^{n}
$$
is a power of 3 .
Note. A power of 3 is a number of the form $3^{a}$ where $a$ is a positive integer.... |
33 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A2. Let $a, b, c$ be positive real numbers such that abc $=1$. Show that
$$
\frac{1}{a^{3}+b c}+\frac{1}{b^{3}+c a}+\frac{1}{c^{3}+a b} \leq \frac{(a b+b c+c a)^{2}}{6}
$$
so
Assistant:
Solution. By the AM-GM in... |
34 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A3. Let $a . b$ c ce positue real numbers such that $a+b+c=a^{2}+b^{2}+c^{2}$. Shou that
$$
\frac{a^{2}}{a^{2}+a b}+\frac{b^{2}}{b^{2}+b c}+\frac{c^{2}}{c^{2}+c a} \geq \frac{a+b+c}{2}
$$
Assistant:
Solution. By ... |
35 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A4. Solve the following equation for $x, y, z \in \mathbb{N}$
$$
\left(1+\frac{x}{y+z}\right)^{2}+\left(1+\frac{y}{z+x}\right)^{2}+\left(1+\frac{z}{x+y}\right)^{2}=\frac{27}{4}
$$
Assistant:
Solution 1. Call $a=1... |
36 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G1. Let $A B C$ be an equilateral triangle, and $P$ a point on the circumcircle of the triangle $A B C$ and distinct from $A, B$ and $C$. If the lines through $P$ and parallel to $B C, C A, A B$ intersect the lines ... |
37 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G4. Let $A B C$ be an acute-angled triangle with circumcircle $\Gamma$, and let $O, H$ be the triangle's circumcenter and orthocenter respectively. Let also $A^{\prime}$ be the point where the angle bisector of angl... |
38 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
C1. Along a round table are arranged 11 cards with the names (all distinct) of the 11 members of the $16^{\text {th }}$ JBMO Problem Selection Committee. The distances between each two consecutive cards are equal. A... |
39 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
C2. $n$ nails nailed on a board are connected by two via a string. Each string is colored in one of $n$ given colors. For any three colors there exist three nails conne.cted by two with strings in these three colors... |
40 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A2. Find the largest possible value of the expression $\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|$ where $x$ is a real number.
Assistant:
Solution. We observe that
$$
\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}... |
41 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
C3. All possible pairs of $n$ apples are weighed and the results are given to us in an arbitrary order. Can we determine the weights of the apples if a. $n=4$, b. $n=5$, c. $n=6$ ?
Assistant:
Solution. a. No. Four... |
42 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G1. Let $A B$ be a diameter of a circle $\omega$ with center $O$ and $O C$ be a radius of $\omega$ which is perpendicular to $A B$. Let $M$ be a point on the line segment $O C$. Let $N$ be the second point of inters... |
43 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G3. Let $D$ be a point on the side $B C$ of an acute triangle $A B C$ such that $\angle B A D=\angle C A O$ where $O$ is the center of the circumcircle $\omega$ of the triangle $A B C$. Let $E$ be the second point o... |
44 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G4. Let $I$ be the incenter and $A B$ the shortest side of a triangle $A B C$. The circle with center $I$ and passing through $C$ intersects the ray $A B$ at the point $P$ and the ray $B A$ at the point $Q$. Let $D$... |
45 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G5. A circle passing through the midpoint $M$ of the side $B C$ and the vertex $A$ of a triangle $A B C$ intersects the sides $A B$ and $A C$ for the second time at the points $P$ and $Q$, respectively. Show that if... |
46 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
N1. Find all positive integere on for othich $1^{2}+2^{2}+\cdots+16^{2}+17$ is a perfort square
Assistant:
Solution. We bave $1^{3}+2^{5}+\cdots+16^{2}=(1+2+\cdots+16)^{2}=5^{2} \quad 17^{0}$ Herese. $11^{3}+2^{3}... |
47 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
N3. Find all ordered pairs $(a, b)$ of positive integers for which the numbers $\frac{a^{3} b-1}{a+1}$ and $\frac{b^{3} a+1}{b-1}$ are positive integers.
Assistant:
Solution. As $a^{3} b-1=b\left(a^{3}+1\right)-(b... |
48 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
N5. Find all ordered triples $(x, y, z)$ of positive integers satisfying the equation
$$
\frac{1}{x^{2}}+\frac{y}{x z}+\frac{1}{z^{2}}=\frac{1}{2013}
$$
Assistant:
Solution. We have $x^{2} z^{2}=2013\left(x^{2}+x... |
49 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## A2 ALB
3) If $x^{3}-3 \sqrt{3} x^{2}+9 x-3 \sqrt{3}-64=0$, find the value of $x^{6}-8 x^{5}+13 x^{4}-5 x^{3}+49 x^{2}-137 x+2015$.
Assistant:
Solution
$x^{3}-3 \sqrt{3} x^{2}+9 x-3 \sqrt{3}-64=0 \Leftrightarro... |
50 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## G1 MNE
Around the triangle $A B C$ the circle is circumscribed, and at the vertex $C$ tangent $t$ to this circle is drawn. The line $p$ which is parallel to this tangent intersects the lines $B C$ and $A C$ at th... |
51 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## G2 MLD
The point $P$ is outside of the circle $\Omega$. Two tangent lines, passing from the point $P$, touch the circle $\Omega$ at the points $A$ and $B$. The median $A M, M \in(B P)$, intersects the circle $\Om... |
52 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G4
CYP
Let $\triangle A B C$ be an acute triangle. The lines $\left(\varepsilon_{1}\right),\left(\varepsilon_{2}\right)$ are perpendicular to $A B$ at the points $A$, $B$, respectively. The perpendicular lines fro... |
53 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
G5 ROU
Let $A B C$ be an acute triangle with $A B \neq A C$. The incircle $\omega$ of the triangle touches the sides $B C, C A$ and $A B$ at $D, E$ and $F$, respectively. The perpendicular line erected at $C$ onto ... |
54 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
NT1 SAU
What is the greatest number of integers that can be selected from a set of 2015 consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
Assistant: ## Solution:
We... |
55 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
NT2 BUL
A positive integer is called a repunit, if it is written only by ones. The repunit with $n$ digits will be denoted by $\underbrace{11 \ldots 1}_{n}$. Prove that:
a) the repunit $\underbrace{11 \ldots 1}_{n... |
56 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## NT3 ALB
a) Show that the product of all differences of possible couples of six given positive integers is divisible by 960 (original from Albania).
b) Show that the product of all differences of possible couples... |
57 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User: ## NT5 MNE
Does there exist positive integers $a, b$ and a prime $p$ such that
$$
a^{3}-b^{3}=4 p^{2} ?
$$
Assistant: ## Solution:
The given equality may be written as
$$
(a-b)\left(a^{2}+a b+b^{2}\right)=4 p^{... |
58 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
C4
GRE
Let $n \geq 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^{2}$ squares of side length 1 . Find the number of parallelograms which have vertices amon... |
59 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A1 Let $a$ be a real positive number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+a x+a^{2}-6=0$ has no solution in the set of the real number.
Assistant: ## Solution
The discriminant of the equation ... |
60 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A2 Prove that $\frac{a^{2}-b c}{2 a^{2}+b c}+\frac{b^{2}-c a}{2 b^{2}+c a}+\frac{c^{2}-a b}{2 c^{2}+a b} \leq 0$ for any real positive numbers $a, b, c$.
Assistant: ## Solution
The inequality rewrites as $\sum \f... |
61 | A conversation between User and Assistant. The user asks a question, and the Assistant solves it. User:
A3 Let $A$ be a set of positive integers containing the number 1 and at least one more element. Given that for any two different elements $m, n$ of $A$ the number $\frac{m+1}{(m+1, n+1)}$ is also an element of $A$, ... |
README.md exists but content is empty.
- Downloads last month
- 5