Found problems: 85335
2010 All-Russian Olympiad, 3
Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$.
2013 USAMO, 1
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.
2020 Tuymaada Olympiad, 2
All non-zero coefficients of the polynomial $f(x)$ equal $1$, while the sum of the coefficients is $20$. Is it possible that thirteen coefficients of $f^2(x)$ equal $9$?
[i](S. Ivanov, K. Kokhas)[/i]
1987 IMO Longlists, 25
Numbers $d(n,m)$, with $m, n$ integers, $0 \leq m \leq n$, are defined by $d(n, 0) = d(n, n) = 0$ for all $n \geq 0$ and
\[md(n,m) = md(n-1,m)+(2n-m)d(n-1,m-1) \text{ for all } 0 < m < n.\]
Prove that all the $d(n,m)$ are integers.
2024 Romania EGMO TST, P2
In a park there are 23 trees $t_0,t_1,\dots,t_{22}$ in a circle and 22 birds $b_1,n_2,\dots,b_{22}.$ Initially, each bird is in a tree. Every minute, the bird $b_i, 1\leqslant i\leqslant 22$ flies from the tree $t_j{}$ to the tree $t_{i+j}$ in clockwise order, indices taken modulo 23. Prove that there exists a moment when at least 6 trees are empty.
1976 IMO Longlists, 10
Show that the reciprocal of any number of the form $2(m^2+m+1)$, where $m$ is a positive integer, can be represented as a sum of consecutive terms in the sequence $(a_j)_{j=1}^{\infty}$
\[ a_j = \frac{1}{j(j + 1)(j + 2)}\]
2019 CCA Math Bonanza, L5.2
Suppose that a planet contains $\left(CCAMATHBONANZA_{71}\right)^{100}$ people ($100$ in decimal), where in base $71$ the digits $A,B,C,\ldots,Z$ represent the decimal numbers $10,11,12,\ldots,35$, respectively. Suppose that one person on this planet is snapping, and each time they snap, at least half of the current population disappears. Estimate the largest number of times that this person can snap without disappearing. An estimate of $E$ earns $2^{1-\frac{1}{200}\left|A-E\right|}$ points, where $A$ is the actual answer.
[i]2019 CCA Math Bonanza Lightning Round #5.2[/i]
2008 iTest Tournament of Champions, 5
Let $c_1,c_2,c_3,\ldots, c_{2008}$ be complex numbers such that \[|c_1|=|c_2|=|c_3|=\cdots=|c_{2008}|=1492,\] and let $S(2008,t)$ be the sum of all products of these $2008$ complex numbers taken $t$ at a time. Let $Q$ be the maximum possible value of \[\left|\dfrac{S(2008,1492)}{S(2008,516)}\right|.\] Find the remainder when $Q$ is divided by $2008$.
2024 Princeton University Math Competition, B1
Let $A=\sqrt{7+2\sqrt{10}} - \sqrt{7-2\sqrt{10}}.$ We can express $A$ as $a\sqrt{b},$ where $a,b$ are integers and $b$ is square-free. Compute $a+b.$
2020 USA TSTST, 3
We say a nondegenerate triangle whose angles have measures $\theta_1$, $\theta_2$, $\theta_3$ is [i]quirky[/i] if there exists integers $r_1,r_2,r_3$, not all zero, such that
\[r_1\theta_1+r_2\theta_2+r_3\theta_3=0.\]
Find all integers $n\ge 3$ for which a triangle with side lengths $n-1,n,n+1$ is quirky.
[i]Evan Chen and Danielle Wang[/i]
1953 AMC 12/AHSME, 37
The base of an isosceles triangle is $ 6$ inches and one of the equal sides is $ 12$ inches. The radius of the circle through the vertices of the triangle is:
$ \textbf{(A)}\ \frac{7\sqrt{15}}{5} \qquad\textbf{(B)}\ 4\sqrt{3} \qquad\textbf{(C)}\ 3\sqrt{5} \qquad\textbf{(D)}\ 6\sqrt{3} \qquad\textbf{(E)}\ \text{none of these}$
1997 Spain Mathematical Olympiad, 5
Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.
1995 IMO Shortlist, 3
The incircle of triangle $ \triangle ABC$ touches the sides $ BC$, $ CA$, $ AB$ at $ D, E, F$ respectively. $ X$ is a point inside triangle of $ \triangle ABC$ such that the incircle of triangle $ \triangle XBC$ touches $ BC$ at $ D$, and touches $ CX$ and $ XB$ at $ Y$ and $ Z$ respectively.
Show that $ E, F, Z, Y$ are concyclic.
1990 Hungary-Israel Binational, 4
A rectangular sheet of paper with integer length sides is given. The sheet is marked with unit squares. Arrows are drawn at each lattice point on the sheet in a way that each arrow is parallel to one of its sides, and the arrows at the boundary of the paper do not point outwards. Prove that there exists at least one pair of neighboring lattice points (horizontally, vertically or diagonally) such that the arrows drawn at these points are in opposite directions.
2015 ASDAN Math Tournament, 8
Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.
2000 India Regional Mathematical Olympiad, 4
All the $7$ digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once , and not divisible by $5$ are arranged in increasing order. Find the $200th$ number in the list.
2003 Alexandru Myller, 3
Let be three elements $ a,b,c $ of a nontrivial, noncommutative ring, that satisfy $ ab=1-c, $ and such that there exists an element $ d $ from the ring such that $ a+cd $ is a unit. Prove that there exists an element $ e $ from the ring such that $ b+ec $ is a unit.
[i]Andrei Nedelcu[/i] and [i] Lucian Ladunca [/i]
2023 VN Math Olympiad For High School Students, Problem 10
Prove that: the polynomial$$(x(x+1)(x+2)(x+3))^{2^{2023}}+1$$is irreducible in $\mathbb{Q}[x].$
2002 Stanford Mathematics Tournament, 2
Solve for all real $x$ that satisfy the equation $4^x=2^x+6$
2015 Harvard-MIT Mathematics Tournament, 9
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in the triangle $BCD$ such that $[XBC]=2[XBD]=4[XCD]$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$. Let $Y$ lie on segment $AX$ such that $2AY=YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.
2024 Bangladesh Mathematical Olympiad, P7
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that\[f\left(\Big \lceil \frac{f(m)}{n} \Big \rceil\right)=\Big \lceil \frac{m}{f(n)} \Big \rceil\]for all $m,n \in \mathbb{N}$.
[i]Proposed by Md. Ashraful Islam Fahim[/i]
2017 Harvard-MIT Mathematics Tournament, 6
A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.
2024 Balkan MO, 4
Let $\mathbb{R}^+ = (0, \infty)$ be the set of all positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0) = 0$ which satisfy the equality $f(f(x) + P(y)) = f(x - y) + 2y$ for all real numbers $x > y > 0$.
[i]Proposed by Sardor Gafforov, Uzbekistan[/i]
2021 SAFEST Olympiad, 1
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
1955 Miklós Schweitzer, 1
[b]1.[/b] Let $a_{1}, a_{2}, \dots , a_{n}$ and $b_{1}, b_{2}, \dots , b_{m}$ be $n+m$ unit vectors in the $r$-dimensional Euclidean space $E_{r} (n,m \leq r)$; let $a_{1}, a_{2}, \dots , a_{n}$ as well as $b_{1}, b_{2}, \dots , b_{m}$ be mutually orthogonal. For any vector $x \in E_{r}$, consider
$Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}$
($(a,b)$ denotes the scalar product of $a$ and $b$). Show that the sequence $(T^{k}x)^{\infty}_{ k =0}$, where $T^{0} x= x$ and $T^{k} x = T(T^{k-1}x)$, is convergent and give a geometrical characterization of how the limit depends on $x$. [b](S. 14)[/b]