Found problems: 85335
2019 Balkan MO Shortlist, N1
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
$$f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q$$
holds for all $p,q\in\mathbb{P}$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
LMT Team Rounds 2021+, 6
An isosceles trapezoid $PQRS$, with $\overline{PQ} = \overline{QR}= \overline{RS}$ and $\angle PQR = 120^o$, is inscribed in the graph of $y = x^2$ such that $QR$ is parallel to the $x$-axis and $R$ is in the first quadrant. The $x$-coordinate of point $R$ can be written as $\frac{\sqrt{A}}{B}$ for positive integers $A$ and $B$ such that $A$ is square-free. Find $1000A +B$.
2024 Turkey EGMO TST, 1
Let $ABC$ be a triangle and its circumcircle be $\omega$. Let $I$ be the incentre of the $ABC$. Let the line $BI$ meet $AC$ at $E$ and $\omega$ at $M$ for the second time. The line $CI$ meet $AB$ at $F$ and $\omega$ at $N$ for the second time. Let the circumcircles of $BFI$ and $CEI$ meet again at point $K$. Prove that the lines $BN$, $CM$, $AK$ are concurrent.
2020 MBMT, 13
How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$?
[i]Proposed by Joshua Hsieh[/i]
2008 Princeton University Math Competition, A3
A sequence $\{a_i\}$ is defined by $a_1 = c$ for some $c > 0$ and $a_{n+1} = a_n + \frac{n}{a_n}$. Prove that $\frac{a_n}{n}$ converges and find its limit.
1999 Mexico National Olympiad, 5
In a quadrilateral $ABCD$ with $AB // CD$, the external bisectors of the angles at $B$ and $C$ meet at $P$, while the external bisectors of the angles at $A$ and $D$ meet at $Q$. Prove that the length of $PQ$ equals the semiperimeter of $ABCD$.
1989 India National Olympiad, 3
Let $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements.
1994 Kurschak Competition, 2
Prove that if we erase $n-3$ diagonals of a regular $n$-gon, then we may still choose $n-3$ of the remaining diagonals such that they don't intersect inside the $n$-gon; but it is possible to erase $n-2$ diagonals such that this statement doesn't hold.
2022-23 IOQM India, 22
A binary sequence is a sequence in which each term is equal to $0$ or $1$. A binary sequence is called $\text{friendly}$ if each term is adjacent to at least on term that is equal to $1$. For example , the sequence $0,1,1,0,0,1,1,1$ is $\text{friendly}$. Let $F_{n}$ denote the number of $\text{friendly}$ binary sequences with $n$ terms. Find the smallest positive integer $n\ge 2$ such that $F_{n}>100$
2011 Dutch IMO TST, 4
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$:
$p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.
2021 Saudi Arabia Training Tests, 28
Find all positive integer $n\ge 3$ such that it is possible to mark the vertices of a regular $n$- gon with the number from 1 to n so that for any three vertices $A, B$ and $C$ with $AB = AC$, the number in $A$ is greater or smaller than both numbers in $B, C$.
2009 IMO Shortlist, 5
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
1981 Poland - Second Round, 4
The given natural numbers are $ k, n $. We inductively define two sequences of numbers $ (a_j) $ and $ (r_j) $ as follows:
Step one: we divide $ k $ by $ n $ and get the quotient $ a_1 $ and the remainder $ r_i $,
step j: we divide $ k+r_{j-1} $ by $ n $ and get the quotient $ a_j $ and the remainder $ r_j $.
Calculate the sum of $ a_1 + \ldots + a_n $.
2017 Korea USCM, 6
Given a positive integer $n$ and a real valued $n\times n$ matrix $A$. $J$ is $n\times n$ matrix with every entry $1$. Suppose $A$ satisfies the following relations.
$$A+A^T = \frac{1}{n} J, \quad AJ = \frac{1}{2} J$$
Show that $A^m-I$ is an invertible matrix for all positive odd integer $m$.
1995 AMC 8, 22
The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?
$\text{(A)}\ 162 \qquad \text{(B)}\ 172 \qquad \text{(C)}\ 173 \qquad \text{(D)}\ 174 \qquad \text{(E)}\ 222$
1995 IberoAmerican, 2
Let $n$ be a positive integer greater than 1. Determine all the collections of real numbers $x_1,\ x_2,\dots,\ x_n\geq1\mbox{ and }x_{n+1}\leq0$ such that the next two conditions hold:
(i) $x_1^{\frac12}+x_2^{\frac32}+\cdots+x_n^{n-\frac12}= nx_{n+1}^\frac12$
(ii) $\frac{x_1+x_2+\cdots+x_n}{n}=x_{n+1}$
1961 All Russian Mathematical Olympiad, 006
a) Points $A$ and $B$ move uniformly and with equal angle speed along the circumferences with $O_a$ and $O_b$ centres (both clockwise). Prove that a vertex $C$ of the equilateral triangle $ABC$ also moves along a certain circumference uniformly.
b) The distance from the point $P$ to the vertices of the equilateral triangle $ABC$ equal $|AP|=2, |BP|=3$. Find the maximal value of $CP$.
1994 Bulgaria National Olympiad, 1
Two circles $k_1(O_1,R)$ and $k_2(O_2,r)$ are given in the plane such that $R \ge \sqrt2 r$ and $$O_1O_2 =\sqrt{R^2 +r^2 - r\sqrt{4R^2 +r^2}}.$$ Let $A$ be an arbitrary point on $k_1$. The tangents from $A$ to $k_2$ touch $k_2$ at $B$ and $C$ and intersect $k_1$ again at $D$ and $E$, respectively. Prove that $BD \cdot CE = r^2$
2014 Iran MO (3rd Round), 5
We say $p(x,y)\in \mathbb{R}\left[x,y\right]$ is [i]good[/i] if for any $y \neq 0$ we have $p(x,y) = p\left(xy,\frac{1}{y}\right)$ . Prove that there are good polynomials $r(x,y) ,s(x,y)\in \mathbb{R}\left[x,y\right]$ such that for any good polynomial $p$ there is a $f(x,y)\in \mathbb{R}\left[x,y\right]$ such that \[f(r(x,y),s(x,y))= p(x,y)\]
[i]Proposed by Mohammad Ahmadi[/i]
1962 IMO Shortlist, 3
Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.
2002 Turkey MO (2nd round), 1
Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition
\[y^2 \equiv x^3 - x \pmod p\]
is exactly $p.$
2015 India PRMO, 8
[b]8.[/b] The fi gure below shows a broken piece of a circular plate made of glass.
[img]https://cdn.artofproblemsolving.com/attachments/7/3/a49f60d803f802c54e2295932b34579514b4fe.png[/img]
$C$ is the midpoint of $AB$, and $D$ is the midpoint of arc $AB$. Given that $AB = 24$ cm and $CD = 6$ cm, what is the radius of the plate in centimetres? (The fi gure is not drawn to scale.)
2022 Turkey Team Selection Test, 4
We have three circles $w_1$, $w_2$ and $\Gamma$ at the same side of line $l$ such that $w_1$ and $w_2$ are tangent to $l$ at $K$ and $L$ and to $\Gamma$ at $M$ and $N$, respectively. We know that $w_1$ and $w_2$ do not intersect and they are not in the same size. A circle passing through $K$ and $L$ intersect $\Gamma$ at $A$ and $B$. Let $R$ and $S$ be the reflections of $M$ and $N$ with respect to $l$. Prove that $A, B, R, S$ are concyclic.
1997 Romania National Olympiad, 1
Let $k$ be an integer number and $P(X)$ be the polynomial $$P(X) = X^{1997}-X^{1995} +X^2-3kX+3k+1$$
Prove that:
a) the polynomial has no integer root;
β) the numbers $P(n)$ and $P(n) + 3$ are relatively prime, for every integer $n$.
2001 Belarusian National Olympiad, 5
In the increasing sequence of positive integers $a_1$, $a_2$,. . . , the number $a_k$ is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence.
(a) Prove that all but finitely terms of the sequence are funny.
(b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers?