Found problems: 85335
Let $ABC$ be an isosceles triangle with $AB=AC$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $AB$ of $\Gamma$ not containing $C$. A point $E$ is on arc $AC$ of $\Gamma$ not containing $B$. If $AD=CE$ prove that $BE$ is parallel to $AD$.
Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
Let $\triangle PQO$ be the unique right isosceles triangle inscribed in the parabola $y = 12x^2$ with $P$ in the first quadrant, right angle at $Q$ in the second quadrant, and $O$ at the vertex $(0, 0)$. Let $\triangle ABV$ be the unique right isosceles triangle inscribed in the parabola $y = x^2/5 + 1$ with $A$ in the first quadrant, right angle at $B$ in the second quadrant, and $V$ at the vertex $(0, 1)$. The $y$-coordinate of $A$ can be uniquely written as $uq^2 + vq + w$, where $q$ is the $x$-coordinate of $Q$ and $u$, $v$, and $w$ are integers. Determine $u + v + w$.
There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle stays in the circle and the other three leave. This continues until one player remains. Everyone has equal rapping ability, i.e. every person has equal probability to win a round. What is the probability that Michael and James end up battling in the same round?
If $a,b,c\in\mathbb N\setminus\{0,1,2,3\}$ then prove:
$$b^2\cdot\sqrt[a]a+c^2\cdot\sqrt[b]b+a^2\cdot\sqrt[c]c\ge48\sqrt2$$
[i]Proposed by Daniel Sitaru[/i]
What is the maximal possible value of $n$ such that no matter how $25$ squares are selected in an infinite chessboard one can find $n$ squares in which none of them share a common corner?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ 11
$
Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have
\[P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)\]
Prove that the polynomial $P(x)$ has at least one real root.
Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if:
[list=i]
[*] the $n^2$ entries are integers from $1$ to $n$;
[*] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and
[*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix.
[/list]
Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.
Construct a right-angled triangle along its two medians, starting from the acute angles.
We say "$s$ grows to $r$" if there exists some integer $n>0$ such that $s^n = r.$ Call a real number $r$ "sparcs" if there are only finitely many real numbers $s$ that grow to $r.$ Find all real numbers that are sparse.
Define sequence $(a_n):a_1=\text{e},a_2=\text{e}^3,\text{e}^{1-k}a_n^{k+2}=a_{n+1}a_{n-1}^{2k}$ for all $n\geq2$, where $k$ is a positive real number. Find $\prod_{i=1}^{2017}a_i$.
Let $n>2$ be a positive integer. Masha writes down $n$ natural numbers along a circle. Next, Taya performs the following operation: Between any two adjacent numbers $a$ and $b$, she writes a divisor of the number $a+b$ greater than $1$, then Taya erases the original numbers and obtains a new set of $n$ numbers along the circle. Can Taya always perform these operations in such a way that after some number of operations, all the numbers are equal?
[i]Proposed by T. Korotchenko[/i]
$ a \minus{} )$ Find all prime $ p$ such that $ \dfrac{7^{p \minus{} 1} \minus{} 1}{p}$ is a perfect square
$ b \minus{} )$ Find all prime $ p$ such that $ \dfrac{11^{p \minus{} 1} \minus{} 1}{p}$ is a perfect square
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\frac{1}{z}=5,$ and $y+\frac{1}{x}=29.$ Then $z+\frac{1}{y}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.
Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$
There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear.
A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.
In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^2$, where $k$ is a constant, find $\rho$ as a function of $r.$ Find also the magnitude of the force of attraction at a point outside the sphere at a distance $r$ from the center.
Let $ f,g:[0,1]\longrightarrow{R} $ be two continuous functions such that $ f(x)g(x)\ge 4x^2, $ for all $ x\in [0,1] . $ Prove that
$$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$
In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.
We are given a balance with two bowls and a number of weights.
(a) Give an example of four integer weights using which one can measure any weight of $1,2,...,40$ grams.
(b) Are there four weights using which one can measure any weight of $1,2,...,50$ grams?
2025 IMO leaders are discussing $100$ problems in a meeting. For each $i = 1, 2,\ldots , 100$, each leader will talk about the $i$-th problem for $i$-th minutes. The chair can assign one leader to talk about a problem of his choice, but he would have to wait for the leader to complete the entire talk of that problem before assigning the next leader and problem. The next leader can be the same leader. The next problem can be a different problem. Each leader’s longest idle time is the longest consecutive time that he is not talking.
Find the minimum positive integer $T$ so that the chair can ensure that the longest idle time for any leader does not exceed $T$.
[i]Proposed by usjl[/i]
Solve the equation $$\left[\frac{x^2+1}{x}\right]-\left[\frac{x}{x^2+1}\right]=3.$$
The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root?
$\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $