Found problems: 85335
1958 Miklós Schweitzer, 4
[b]4.[/b] Let $P_1 P_2 P_3 P_4 P_5 P_6$ be a convex hexagon. Denote by $T$ its area and by $t$ the area of the triangle $Q_1 Q_2 Q_3$, where $Q_1,Q_2$ and $Q_3$ are the midpoints of $P_1P_4,P_2P_5,P_3P_6$ respectively. Prove that $t<\frac{1}{4}T$. [b](G. 3)[/b]
2010 Saint Petersburg Mathematical Olympiad, 1
Chess king is standing in some square of chessboard. Every sunday it is moved to one square by diagonal, and every another day it is moved to one square by horisontal or vertical. What maximal numbers of moves can be made ?
2012 EGMO, 4
A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$.
Does there exist a sum-full zero-sum-free set of integers?
[i]Romania (Dan Schwarz)[/i]
2010 Tournament Of Towns, 1
Alex has a piece of cheese. He chooses a positive number $a\neq 1$ and cuts the piece into several pieces one by one. Every time he chooses a piece and cuts it in the same ratio $1:a.$ His goal is to divide the cheese into two piles of equal masses. Can he do it?
2011 Croatia Team Selection Test, 1
We define a sequence $a_n$ so that $a_0=1$ and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \]
for all postive integers $n$.
Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.
2022 Costa Rica - Final Round, 1
Let $\Gamma$ be a circle with center $O$. Consider the points $A$, $B$, $C$, $D$, $E$ and $F$ in $\Gamma$, with $D$ and $E$ in the (minor) arc $BC$ and $C$ in the (minor) arc $EF$, such that $DEFO$ is a rhombus and $\vartriangle ABC$ It is equilateral. Show that $\overleftrightarrow{BD}$ and $\overleftrightarrow{CE}$ are perpendicular.
2021 Saudi Arabia Training Tests, 34
Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.
2018 Switzerland - Final Round, 2
Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take:
$$\frac{a}{gcd\,\,(a + b, a - c)}
+
\frac{b}{gcd\,\,(b + c, b - a)}
+
\frac{c}{gcd\,\,(c + a, c - b)}.$$
.
Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.
1991 Brazil National Olympiad, 2
$P$ is a point inside the triangle $ABC$. The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$. Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$, and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$. Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$.
2007 Princeton University Math Competition, 7
Positive reals $p$ and $q$ are such that the graph of $y = x^2 - 2px + q$ does not intersect the $x$-axis. Find $q$ if there is a unique pair of points $A, B$ on the graph with $AB$ parallel to the $x$-axis and $\angle AOB = \frac{\pi}{2}$, where $O$ is the origin.
2006 Romania National Olympiad, 2
Prove that for all $\displaystyle a,b \in \left( 0 ,\frac{\pi}{4} \right)$ and $\displaystyle n \in \mathbb N^\ast$ we have \[ \frac{\sin^n a + \sin^n b}{\left( \sin a + \sin b \right)^n} \geq \frac{\sin^n 2a + \sin^n 2b}{\left( \sin 2a + \sin 2b \right)^n} . \]
2020 Canadian Mathematical Olympiad Qualification, 6
In convex pentagon $ABCDE, AC$ is parallel to $DE, AB$ is perpendicular to $AE$, and $BC$ is perpendicular to $CD$. If $H$ is the orthocentre of triangle $ABC$ and $M$ is the midpoint of segment $DE$, prove that $AD, CE$ and $HM$ are concurrent.
2005 Belarusian National Olympiad, 3
Solve in positive integers $a>b$:
$$(a-b)^{ab}=a^bb^a$$
2024 Ukraine National Mathematical Olympiad, Problem 3
Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that:
$$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$
Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$.
[i]Proposed by Anton Trygub[/i]
2022 VN Math Olympiad For High School Students, Problem 6
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Let $G$ be the centroid of $\triangle ABC$.
Prove that: the distances from $G$ to the perpendicular bisectors of $TA, TB, TC$ are the same.
2021 Polish MO Finals, 2
Let $n$ be an integer. For pair of integers $0 \leq i,$ $j\leq n$ there exist real number $f(i,j)$ such that:
1) $ f(i,i)=0$ for all integers $0\leq i \leq n$
2) $0\leq f(i,l) \leq 2\max \{ f(i,j), f(j,k), f(k,l) \}$ for all integers $i$, $j$, $k$, $l$ satisfying $0\leq i\leq j\leq k\leq l\leq n$.
Prove that $$f(0,n) \leq 2\sum_{k=1}^{n}f(k-1,k)$$
2017 VJIMC, 3
Let $n \ge 2$ be an integer. Consider the system of equations
\begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align}
1. Prove that $(1)$ has infinitely many real solutions $(x_1,\dotsc,x_n)$ such that the numbers $x_1,\dotsc,x_n$ are distinct.
2. Prove that every solution of $(1)$, such that the numbers $x_1,\dotsc,x_n$ are not all equal, satisfies $\vert x_1x_2\cdots x_n\vert=2^{n/2}$.
2016 Cono Sur Olympiad, 1
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.
[b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
2021 Mexico National Olympiad, 5
If $n=\overline{a_1a_2\cdots a_{k-1}a_k}$, be $s(n)$ such that
. If $k$ is even, $s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k}$
. If $k$ is odd, $s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k}$
For example $s(123)=1+23=24$ and $s(2021)=20+21=41$
Be $n$ is $digital$ if $s(n)$ is a divisor of $n$. Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is $digital$.
2023 Switzerland Team Selection Test, 5
The Tokyo Metro system is one of the most efficient in the world. There is some odd positive integer $k$ such that each metro line passes through exactly $k$ stations, and each station is serviced by exactly $k$ metro lines. One can get from any station to any otherstation using only one metro line - but this connection is unique. Furthermore, any two metro lines must share exactly one station. David is planning an excursion for the IMO team, and wants to visit a set $S$ of $k$ stations. He remarks that no three of the stationsin $S$ are on a common metro line. Show that there is some station not in $S$, which is connected to every station in $S$ by a different metro line.
2006 Princeton University Math Competition, 6
Evaluate the sum $$\sum_{k=0}^{r} {r \choose k}{{12-r} \choose {6-k}} $$
2022 MIG, 13
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave?
$\textbf{(A) }23\qquad\textbf{(B) }27\qquad\textbf{(C) }29\qquad\textbf{(D) }31\qquad\textbf{(E) }33$
2011 ELMO Shortlist, 3
Let $ABC$ be a triangle. Draw circles $\omega_A$, $\omega_B$, and $\omega_C$ such that $\omega_A$ is tangent to $AB$ and $AC$, and $\omega_B$ and $\omega_C$ are defined similarly. Let $P_A$ be the insimilicenter of $\omega_B$ and $\omega_C$. Define $P_B$ and $P_C$ similarly. Prove that $AP_A$, $BP_B$, and $CP_C$ are concurrent.
[i]Tom Lu.[/i]
2013 Today's Calculation Of Integral, 897
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
2025 NCMO, 1
A collection of $n$ positive numbers, where repeats are allowed, adds to $500$. They can be split into $20$ groups each adding to $25$, and can also be split into $25$ groups each adding to $20$. (A group is allowed to contain any amount of integers, even just one integer.) What is the least possible value of $n$?
[i]Aaron Wang[/i]