Found problems: 85335
2022 IMAR Test, 3
Given is a parallelogram $XYZT$, and the variable points $A, B, C, D$ lie on the sides $XY, XT, TZ, ZY$ respectively, so that $ABCD$ is cyclic with circumcenter $O$, $AC \parallel XT$, and $BD \parallel XY$. Let $P$ be the intersection point of the lines $AD$ and $BC$, and let $Q$ be the intersection of the lines $AB$ and $CD$. Prove that the circle $(POQ)$ passes through a fixed point.
2024 Assara - South Russian Girl's MO, 7
There is a chip in one of the squares on the checkered board. In one move, she can move either $1$ square to the right, or diagonally $1$ to the left and $1$ up, or $1$ to the left and $3$ down (see Fig.). The chip made $n$ moves and returned to the starting square. Prove that a) $n$ is divisible by $2$, b) $n$ is divisible by $8$.
[i]K.A.Sukhov[/i]
2014 Bosnia Herzegovina Team Selection Test, 1
Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$
2021 MIG, 2
Solve for $x$ if $20x + 21 = 121$.
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }7$
2017 VJIMC, 1
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function satisfying
\[f(x+2y)=2f(x)f(y)\]
for every $x,y \in \mathbb{R}$. Prove that $f$ is constant.
2005 AMC 12/AHSME, 24
All three vertices of an equilateral triangle are on the parabola $ y \equal{} x^2$, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum of $ m/n$, where $ m$ and $ n$ are relatively prime positive integers. What is the value of $ m \plus{} n$?
$ \textbf{(A)}\ 14\qquad
\textbf{(B)}\ 15\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 17\qquad
\textbf{(E)}\ 18$
2016 Harvard-MIT Mathematics Tournament, 2
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
2023 Peru MO (ONEM), 4
Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.
2009 Korea Junior Math Olympiad, 7
There are $3$ students from Korea, China, and Japan, so total of $9$ students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array $A$ can become array $B$ by rotation, these two arrays are considered equal.
2014 BAMO, 5
A chess tournament took place between $2n+1$ players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly $k$ games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than $n-\sqrt{2k}$ and no more than $n+\sqrt{2k}$ games.
2025 Bulgarian Spring Mathematical Competition, 11.4
We call two non-constant polynomials [i]friendly[/i] if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials \( P(x), Q(x) \) and a constant \( C \in \mathbb{R}, C \neq 0 \), it is given that \( P(x) + C \) and \( Q(x) + C \) are also friendly polynomials. Prove that \( P(x) \equiv Q(x) \).
2020 Canadian Mathematical Olympiad Qualification, 7
Let $a, b, c$ be positive real numbers with $ab + bc + ac = abc$. Prove that
$$\frac{bc}{a^{a+1}} +\frac{ac}{b^{b+1 }}+\frac{ab}{c^{c+1}} \ge \frac13$$
2005 Canada National Olympiad, 5
Let's say that an ordered triple of positive integers $(a,b,c)$ is [i]$n$-powerful[/i] if $a\le b\le c,\gcd (a,b,c)=1$ and $a^n+b^n+c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is $5$-powerful.
$a)$ Determine all ordered triples (if any) which are $n$-powerful for all $n\ge 1$.
$b)$ Determine all ordered triples (if any) which are $2004$-powerful and $2005$-powerful, but not $2007$-powerful.
2016 Japan MO Preliminary, 5
Let $ABCD$ be a quadrilateral with $AC=20$, $AD=16$. We take point $P$ on segment $CD$ so that triangle $ABP$ and $ACD$ are congruent. If the area of triangle $APD$ is $28$, find the area of triangle $BCP$. Note that $XY$ expresses the length of segment $XY$.
2021 LMT Spring, A13
In a round-robin tournament, where any two players play each other exactly once, the fact holds that among every three students $A$, $B$, and $C$, one of the students beats the other two. Given that there are six players in the tournament and Aidan beats Zach but loses to Andrew, find how many ways there are for the tournament to play out. Note: The order in which the matches take place does not matter.
[i]Proposed by Kevin Zhao[/i]
2020 Simon Marais Mathematics Competition, B4
[i]The following problem is open in the sense that no solution is currently known to part (b).[/i]
Let $n\geq 2$ be an integer, and $P_n$ be a regular polygon with $n^2-n+1$ vertices.
We say that $n$ is $\emph{taut}$ if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct.
(a) show that if $n-1$ is prime then $n$ is taut.
(b) Which integers $n\geq 2$ are taut?
2002 HKIMO Preliminary Selection Contest, 2
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer.
PEN H Problems, 9
Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.
2019 Peru EGMO TST, 3
For a finite set $A$ of integers, define $s(A)$ as the number of values obtained by adding any two elements of $A$, not necessarily different. Analogously, define $r (A)$ as the number of values obtained by subtracting any two elements of $A$, not necessarily different.
For example, if $A = \{3,1,-1\}$
$\bullet$ The values obtained by adding any two elements of $A$ are $\{6,4,2,0,-2\}$ and so $s (A) = 5$.
$\bullet$ The values obtained by subtracting any two elements of $A$ are $\{4,2,0,-2,-4\}$ and as $r (A) = 5$.
Prove that for each positive integer $n$ there is a finite set $A$ of integers such that $r (A) \ge n s (A)$.
MOAA Team Rounds, 2019.10
Let $S$ be the set of all four digit palindromes (a palindrome is a number that reads the same forwards and backwards). The average value of $|m - n|$ over all ordered pairs $(m, n)$, where $m$ and $n$ are (not necessarily distinct) elements of $S$, is equal to $p/q$ , for relatively prime positive integers $p$ and $q$. Find $p + q$.
2014 Postal Coaching, 1
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.Determine those numbers $n$ for which $d(n^3)=5d(n)$.
2022 Federal Competition For Advanced Students, P2, 6
(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$.
(b) Show that a corresponding decomposition into $30$ squares is also possible.
[i](Walther Janous)[/i]
2015 Indonesia MO Shortlist, N6
Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.
2014 Tuymaada Olympiad, 1
Given are three different primes. What maximum number of these primes can divide their sum?
[i](A. Golovanov)[/i]
2008 Regional Olympiad of Mexico Center Zone, 2
Let $ABC$ be a triangle with incenter $I $, the line $AI$ intersects $BC$ at $ L$ and the circumcircle of $ABC$ at $L'$. Show that the triangles $BLI$ and $L'IB$ are similar if and only if $AC = AB + BL$.