Found problems: 85335
1995 IberoAmerican, 2
The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. Let the line $AD$ intersect this incircle of triangle $ABC$ at a point $X$ (apart from $D$). Assume that this point $X$ is the midpoint of the segment $AD$, this means, $AX = XD$. Let the line $BX$ meet the incircle of triangle $ABC$ at a point $Y$ (apart from $X$), and let the line $CX$ meet the incircle of triangle $ABC$ at a point $Z$ (apart from $X$). Show that $EY = FZ$.
2010 LMT, 15
Al is bored of Rock Paper Scissors, and wants to invent a new game: $Z-Y-X-W-V.$ Two players, each choose to play either $Z, Y, X, W,$ or $V.$ If they play the same thing, the result is a tie. However, Al must come up with a ’pecking order’, that is, he must decide which plays beat which. For each of the $10$ pairs of distinct plays that the two players can make, Al randomly decides a winner. For example, he could decide that $W$ beats $Y$ and that $Z$ beats $X,$ etc. What is the probability that after Al makes all of these $10$ choices, the game is balanced, that is, playing each letter results in an equal probability of winning?
2008 Germany Team Selection Test, 3
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$
[i]Author: Gerhard Wöginger, Netherlands[/i]
2019 Sharygin Geometry Olympiad, 6
Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic
1997 Belarusian National Olympiad, 4
A triangle $A_1B_1C_1$ is a parallel projection of a triangle $ABC$ in space. The parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\vartriangle ABC$ respectively are drawn. Using a ruler and compass, construct a parallel projection of :
(a) the orthocenter,
(b) the incenter of $\vartriangle ABC$.
2025 239 Open Mathematical Olympiad, 8
The incircle of a right triangle $ABC$ touches its hypotenuse $BC$ at point $D$. The line $AD$ intersects the circumscribed circle at point $X$. Prove that $ |BX-CX| \geqslant |AD - DX|$.
2010 Contests, 2
Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.
2001 Chile National Olympiad, 4
Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.
1968 Polish MO Finals, 4
Given an integer $n > 2$, give an example of a set of $n$ mutually different numbers $a_1,...,a_n$ for which the set of their pairwise sums $a_i + a_j$ ($i \ne j$) contains as few different numbers as possible; also give an example of a set of n different numbers $b_1,...,b_n$ for which the set of their pairwise sums $b_i+b_j$ ($i \ne j$) contains as many different numbers as possible;
2020-2021 OMMC, 8
Triangle $ABC$ has circumcircle $\omega$. The angle bisectors of $\angle A$ and $\angle B$ intersect $\omega$ at points $D$ and $E$ respectively. $DE$ intersects $BC$ and $AC$ at $X$ and $Y$ respectively. Given $DX = 7,$ $XY = 8$ and $YE = 9,$ the area of $\triangle ABC$ can be written as $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are positive integers, $\gcd(a,c) = 1,$ and $b$ is square free. Find $a+b+c.$
2024 Australian Mathematical Olympiad, P7
Let $ABCD$ be a square and let $P$ be a point on side $AB$. The point $Q$ lies outside the square such that $\angle ABQ = \angle ADP$ and $\angle AQB = 90^{\circ}$. The point $R$ lies on the side $BC$ such that $\angle BAR = \angle ADQ$. Prove that the lines $AR, CQ$ and $DP$ pass through a common point.
2011 Balkan MO Shortlist, A4
Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$. Prove, that
\begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}
2021 Sharygin Geometry Olympiad, 10-11.7
Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.
2012 South East Mathematical Olympiad, 1
Find a triple $(l, m, n)$ of positive integers $(1<l<m<n)$, such that $\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k$ form a geometric sequence in order.
2014 Taiwan TST Round 1, 6
In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most $100$ cities at distance exactly three from it. Prove that there is no city such that more than $2550$ other cities have distance exactly four from it.
2021 Peru IMO TST, P3
Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions:
(i) $f(f(x))=x^2$ for any $x\geq 1$;
(ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$.
1. Prove that $x<f(x)<x^2$ for any $x\geq 1$.
2. Prove that there exists a function $f$ satisfies the above two conditions and the following one:
(iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.
2011 Indonesia TST, 1
Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that
$$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$
MathLinks Contest 3rd, 1
Let $a, b, c$ be positive reals. Prove that $$\sqrt{abc}(\sqrt{a} +\sqrt{b} +\sqrt{c}) + (a + b + c)^2 \ge 4 \sqrt{3abc(a + b + c)}.$$
2016 Online Math Open Problems, 6
For a positive integer $n$, define $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$. Find the positive integer $k$ for which $7?9?=5?k?$.
[i]Proposed by Tristan Shin[/i]
2021 HMNT, 7
De ne the function $f : R \to R$ by $$f(x) =\begin{cases}
\dfrac{1}{x^2+\sqrt{x^4+2x}}\,\,\,
\text{if} \,\,\,x \notin (- \sqrt[3]{2}, 0] \\
\,\,\, 0 \,\,\,, \,\,\, \text{otherwise}
\end{cases}$$
The sum of all real numbers $x$ for which $f^{10}(x) = 1$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a, b,c, d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d.$
(Here, $f^n(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^3(x) = f(f(f(x)))$.)
2009 IMAC Arhimede, 4
Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ .
Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .
2013 Korea National Olympiad, 6
Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.
2018 Sharygin Geometry Olympiad, 8
Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.
2021 Macedonian Team Selection Test, Problem 1
Let $k\geq 2$ be a natural number. Suppose that $a_1, a_2, \dots a_{2021}$ is a monotone decreasing sequence of non-negative numbers such that \[\sum_{i=n}^{2021}a_i\leq ka_n\] for all $n=1,2,\dots 2021$. Prove that $a_{2021}\leq 4(1-\frac{1}{k})^{2021}a_1$.
1997 Taiwan National Olympiad, 6
Show that every number of the form $2^{p}3^{q}$ , where $p,q$ are nonnegative integers, divides some number of the form $a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}$, where $a_{2i}\in\{1,2,...,9\}$