Found problems: 85335
2020 Kosovo National Mathematical Olympiad, 2
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.
2015 İberoAmerican, 3
Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$:
$s_n = \alpha^n + \beta^n$
$t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$
Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.
2009 Balkan MO Shortlist, G6
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
2018 Junior Balkan Team Selection Tests - Romania, 1
Prove that the equation $x^2+y^2+z^2 = x+y+z+1$ has no rational solutions.
2024 IMC, 5
Let $n>d$ be positive integers. Choose $n$ independent, uniformly distributed random points $x_1,\dots,x_n$ in the unit ball $B \subset \mathbb{R}^d$ centered at the origin. For a point $p \in B$ denote by $f(p)$ the probability that the convex hull of $x_1,\dots,x_n$ contains $p$. Prove that if $p,q \in B$ and the distance of $p$ from the origin is smaller than the distance of $q$ from the origin, then $f(p) \ge f(q)$.
1998 Korea - Final Round, 2
Let $D$,$E$,$F$ be points on the sides $BC$,$CA$,$AB$ respectively of a triangle $ABC$. Lines $AD$,$BE$,$CF$ intersect the circumcircle of $ABC$ again at $P$,$Q$,$R$, respectively.Show that:
\[\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9\]
and find the cases of equality.
1994 AIME Problems, 10
In triangle $ABC,$ angle $C$ is a right angle and the altitude from $C$ meets $\overline{AB}$ at $D.$ The lengths of the sides of $\triangle ABC$ are integers, $BD=29^3,$ and $\cos B=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
MathLinks Contest 4th, 3.3
Let $ABC$ be a triangle, and let $C$ be its circumcircle. Let $T$ be the circle tangent to $AB, AC$ and $C$ internally in the points $F, E$ and $D$ respectively. Let $P, Q$ be the intersection points between the line $EF$ and the lines $DB$ and $DC$ respectively. Prove that if $DP = DQ$ then the triangle $ABC$ is isosceles.
2005 Today's Calculation Of Integral, 62
For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$.
Evaluate
\[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]
2011 Dutch BxMO TST, 4
Let $n \ge 2$ be an integer. Let $a$ be the greatest positive integer such that $2^a | 5^n - 3^n$.
Let $b$ be the greatest positive integer such that $2^b \le n$. Prove that $a \le b + 3$.
2015 Harvard-MIT Mathematics Tournament, 9
Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$.
2002 China Team Selection Test, 1
Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that
\begin{align*}
P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right),
\end{align*}
where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]
2017 Putnam, A2
Let $Q_0(x)=1$, $Q_1(x)=x,$ and
\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]
for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
1987 Polish MO Finals, 5
Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).
2020 USMCA, 16
How many paths from $(0, 0)$ to $(2020, 2020)$, consisting of unit steps up and to the right, pass through at most one point with both coordinates even, other than $(0,0)$ and $(2020,2020)$?
2000 All-Russian Olympiad, 3
In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.
2005 Taiwan TST Round 2, 1
Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$.
This is much harder than the problems we had in the 1st TST...
2009 District Olympiad, 4
Fin the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that:
$$ \frac{f(x+y)+f(x)}{2x+f(y)} =\frac{2y+f(x)}{f(x+y)+f(y)} ,\quad\forall x,y\in\mathbb{N} . $$
2000 Harvard-MIT Mathematics Tournament, 23
How many $7$-digit numbers with distinct digits can be made that are divisible by $3$?
2007 Poland - Second Round, 3
An equilateral triangle with side $n$ is built with $n^{2}$ [i]plates[/i] - equilateral triangles with side $1$. Each plate has one side black, and the other side white. We name [i]the move[/i] the following operation: we choose a plate $P$, which has common sides with at least two plates, whose visible side is the same color as the visible side of $P$. Then, we turn over plate $P$.
For any $n\geq 2$ decide whether there exists an innitial configuration of plates permitting for an infinite sequence of moves.
2015 Grand Duchy of Lithuania, 3
A table consists of $17 \times 17$ squares. In each square one positive integer from $1$ to $17$ is written, every such number is written in exactly $17$ squares. Prove that there is a row or a column of the table that contains at least $5$ different numbers.
1957 Moscow Mathematical Olympiad, 355
a) A student takes a subway to an Olympiad, pays one ruble and gets his change. Prove that if he takes a tram (street car) on his way home, he will have enough coins to pay the fare without change.
b) A student is going to a club. (S)he takes a tram, pays one ruble and gets the change. Prove that on the way back by a tram (s)he will be able to pay the fare without any need to change.
Note: In $1957$, the price of a subway ticket was $50$ kopeks, that of a tram ticket $30$ kopeks, the denominations of the coins were $1, 2, 3, 5, 10, 15$, and $20$ kopeks. ($1$ rouble = $100$ kopeks.)
1954 Poland - Second Round, 5
Given points $ A $, $ B $, $ C $ and $ D $ that do not lie in the same plane. Draw a plane through the point $ A $ such that the orthogonal projection of the quadrilateral $ ABCD $ on this plane is a parallelogram.
Gheorghe Țițeica 2024, P1
Let $E(x,y)=\frac{(1+x)(1+y)(1+xy)}{(1+x^2)(1+y^2)}$. Find the minimum and maximum value of $E$ on $\mathbb{R}^2$.
[i]Dorel Miheț[/i]
1996 Vietnam National Olympiad, 3
Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying
1) There exist s, t such that 1<=s<t<=k and a_s>a_t.
2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2.
P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...