Found problems: 85335
1985 Balkan MO, 2
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that
$\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$.
Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$
2001 Tournament Of Towns, 4
Two persons play a game on a board divided into $3\times 100$ squares. They move in turn: the first places tiles of size $1\times2$ lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?
2020 Baltic Way, 18
Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that
\begin{align*}
x^2+y^2+z^2 &\equiv 0 \pmod n;\\
xyz &\equiv k \pmod n.
\end{align*}
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
2012 IMO, 1
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$
(The [i]excircle[/i] of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)
[i]Proposed by Evangelos Psychas, Greece[/i]
2007 Pre-Preparation Course Examination, 5
Prove that the equation
\[y^3=x^2+5\]
doesn't have any solutions in $Z$.
2018 IMO Shortlist, G5
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
2012 Tournament of Towns, 4
Given a triangle $ABC$. Suppose I is its incentre, and $X, Y, Z$ are the incentres of triangles $AIB, BIC$ and $AIC$ respectively. The incentre of triangle $XYZ$ coincides with $I$. Is it necessarily true that triangle $ABC$ is regular?
2011 Switzerland - Final Round, 10
On each square of an $n\times n$-chessboard, there are two bugs. In a move, each bug moves to a (vertically of horizontally) adjacent square. Bugs from the same square always move to different squares. Determine the maximal number of free squares that can occur after one move.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 10)[/i]
2015 Princeton University Math Competition, A6/B7
We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]
2024 JHMT HS, 4
Let $x$ be a real number satisfying
\[ \sqrt[3]{125-x^3}-\sqrt[3]{27-x^3}=7. \]
Compute $|\sqrt[3]{125-x^3}+\sqrt[3]{27-x^3}|$.
2021 Belarusian National Olympiad, 11.2
Points $A_1,B_1,C_1$ lie on sides $BC, CA, AB$ of an acute-angled triangle, respectively. Denote by $P, Q, R$ the intersections of $BB_1$ and $CC_1$, $CC_1$ and $AA_1$, $AA_1$ and $BB_1$. If triangle $PQR$ is similar to $ABC$ and $\angle AB_1C_1 = \angle BC_1A_1 = \angle CA_1B_1$, prove that $ABC$ is equilateral.
2011 Czech and Slovak Olympiad III A, 4
Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.
2006 Purple Comet Problems, 6
The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.
2016 Iran Team Selection Test, 5
Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.
2018 PUMaC Live Round, Calculus 1
Freddy the king of flavortext has an infinite chest of coins. For each number \(p\) in the interval \([0, 1]\), Freddy has a coin that has probability \(p\) of coming up heads. Jenny the Joyous pulls out a random coin from the chest and flips it 10 times, and it comes up heads every time. She then flips the coin again. If the probability that the coin comes up heads on this 11th flip is \(\frac{p}{q}\) for two integers \(p, q\), find \(p + q\).
Note: flavortext is made up
2004 Singapore Team Selection Test, 1
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by
$x_i= 2^i$ if $0 \leq i \leq 2003$
$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
1972 All Soviet Union Mathematical Olympiad, 165
Let $O$ be the intersection point of the diagonals of the convex quadrangle $ABCD$ . Prove that the line drawn through the points of intersection of the medians of triangles $AOB$ and $COD$ is orthogonal to the line drawn through the points of intersection of the heights of triangles $BOC$ and $AOD$ .
2012 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$
2003 Mexico National Olympiad, 1
Find all positive integers with two or more digits such that if we insert a $0$ between the units and tens digits we get a multiple of the original number.
2007 Tournament Of Towns, 2
Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number
[list][b]a)[/b] $x^2$;
[b]b)[/b] $xy$?[/list]
2017 VJIMC, 4
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
2018 Turkey Team Selection Test, 3
A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.
2017 Saudi Arabia Pre-TST + Training Tests, 9
Let $ABC$ be a triangle inscribed in circle $(O)$, with its altitudes $BH_b, CH_c$ intersect at orthocenter $H$ ($H_b \in AC$, $H_c \in AB$). $H_bH_c$ meets $BC$ at $P$. Let $N$ be the midpoint of $AH, L$ be the orthogonal projection of $O$ on the symmedian with respect to angle $A$ of triangle $ABC$. Prove that $\angle NLP = 90^o$.
2017 Junior Balkan Team Selection Tests - Moldova, Problem 6
Let $a,b$ and $c$ be real numbers such that $|a+b|+|b+c|+|c+a|=8.$
Find the maximum and minimum value of the expression $P=a^2+b^2+c^2.$
2025 6th Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABCDE$ be a pentagon such that $\angle DCB < 90^{\circ} < \angle EDC$. The circle with diameter $BD$ intersects the line $BC$ again at $F$, and the circle with diameter $DE$ intersects the line $CE$ again at $G$. Prove that the second intersection ($\neq D$) of the circumcircle of $\triangle DFG$ and the circle with diameter $AD$ lies on $AC$.
Proposed by [i]Petar Filipovski[/i]