Found problems: 85335
2023-IMOC, A1
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $n$, there exists an unique positive integer $k$, satisfying $f^k(n)\leq n+k+1$.
2006 Kyiv Mathematical Festival, 5
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
All the positive integers from 1 till 1000 are written on the blackboard in some order and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1\le a_1<a_2<\ldots<a_{10}\le1000$ in collection then it is allowed to arrange in increasing order the numbers at places $a_1, a_2, \ldots, a_{10},$ counting from left to right. What is the smallest amount of cards in the collection which enables us to arrange in
increasing order all the numbers for any initial arrangement of them?
1975 Miklós Schweitzer, 8
Prove that if \[ \sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)\] holds for a sequence $ \{a_n \}$ of nonnegative real numbers with some positive integer $ N$, then $ \alpha_{i+p} \geq p \alpha_i$ for $ i,p=1,2,...,$ where \[ \alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .\]
[i]L. Leindler[/i]
2013 Greece JBMO TST, 3
If $p$ is a prime positive integer and $x,y$ are positive integers,
find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1)
If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).
2022 Iran Team Selection Test, 4
Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$, $BCP$ and $OMN$, respectively. The intersection point of $\omega_1$ and $\omega_3$, which is not on the arc $APD$ of $\omega_1$, is $E$ and the intersection point of $\omega_2$ and $\omega_3$, which is not on the arc $BPC$ of $\omega_2$, is $F$. Prove that $OF=OE$.
Proposed by Seyed Amirparsa Hosseini Nayeri
2010 Baltic Way, 11
Let $ABCD$ be a square and let $S$ be the point of intersection of its diagonals $AC$ and $BD$. Two circles $k,k'$ go through $A,C$ and $B,D$; respectively. Furthermore, $k$ and $k'$ intersect in exactly two different points $P$ and $Q$. Prove that $S$ lies on $PQ$.
2021 CMIMC, 9
Let $ABC$ be a triangle with circumcenter $O$. Additionally, $\angle BAC=20^\circ$ and $\angle BCA = 70^\circ$. Let $D, E$ be points on side $AC$ such that $BO$ bisects $\angle ABD$ and $BE$ bisects $\angle CBD$. If $P$ and $Q$ are points on line $BC$ such that $DP$ and $EQ$ are perpendicular to $AC$, what is $\angle PAQ$?
[i]Proposed by Daniel Li[/i]
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]
2016 IFYM, Sozopol, 3
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.
2017 Latvia Baltic Way TST, 1
Prove that for all real $x > 0$ holds the inequality $$\sqrt{\frac{1}{3x+1}}+\sqrt{\frac{x}{x+3}}\ge 1.$$
For what values of $x$ does the equality hold?
2008 SDMO (Middle School), 5
For a positive integer $n$, let $f\left(n\right)$ be the sum of the first $n$ terms of the sequence $$0,1,1,2,2,3,3,4,4,\ldots,r,r,r+1,r+1,\ldots.$$ For example, $f\left(5\right)=0+1+1+2+2=6$.
(a) Find a formula for $f\left(n\right)$.
(b) Prove that $f\left(s+t\right)-f\left(s-t\right)=st$ for all positive integers $s$ and $t$, where $s>t$.
2022 Romania Team Selection Test, 3
Let $ABC$ be an acute triangle such that $AB < AC$. Let $\omega$ be the circumcircle of $ABC$
and assume that the tangent to $\omega$ at $A$ intersects the line $BC$ at $D$. Let $\Omega$ be the circle with
center $D$ and radius $AD$. Denote by $E$ the second intersection point of $\omega$ and $\Omega$. Let $M$ be the
midpoint of $BC$. If the line $BE$ meets $\Omega$ again at $X$, and the line $CX$ meets $\Omega$ for the second
time at $Y$, show that $A, Y$, and $M$ are collinear.
[i]Proposed by Nikola Velov, North Macedonia[/i]
1977 IMO Shortlist, 6
Let $n$ be a positive integer. How many integer solutions $(i, j, k, l) , \ 1 \leq i, j, k, l \leq n$, does the following system of inequalities have:
\[1 \leq -j + k + l \leq n\]\[1 \leq i - k + l \leq n\]\[1 \leq i - j + l \leq n\]\[1 \leq i + j - k \leq n \ ?\]
2004 Irish Math Olympiad, 2
$A$ and $B$ are distinct points on a circle $T$. $C$ is a point distinct from $B$ such that $|AB|=|AC|$, and such that $BC$ is tangent to $T$ at $B$. Suppose that the bisector of $\angle ABC$ meets $AC$ at a point $D$ inside $T$. Show that $\angle ABC>72^\circ$.
1993 ITAMO, 4
Let $P$ be a point in the plane of a triangle $ABC$, different from its circumcenter. Prove that the triangle whose vertices are the projections of $P$ on the perpendicular bisectors of the sides of $ABC$, is similar to $ABC$.
2012 JBMO TST - Turkey, 4
Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.
1962 Bulgaria National Olympiad, Problem 1
It is given the expression $y=\frac{x^2-2x+1}{x^2-2x+2}$, where $x$ is a variable. Prove that:
(a) if $x_1$ and $x_2$ are two values of $x$, the $y_1$ and $y_2$ are the respective values of $y$ only if $x_1<x_2$, $y_1<y_2$;
(b) when $x$ is varying $y$ attains all possible values for which $0\le y<1$.
2024 Thailand Mathematical Olympiad, 5
Let $ABC$ be a scalene triangle. Let $H$ be its orthocenter and $D$ is a foot of altitude from $A$ to $BC$. Also, let $S$ and $T$ be points on the circumcircle of triangle $ABC$ such that $\angle BSH=\angle CTH=90^{\circ}$. Given that $AH=2HD$, prove that $D,S,T$ are collinear.
1974 Czech and Slovak Olympiad III A, 4
Let $\mathcal M$ be the set of all polynomial functions $f$ of degree at most 3 such that \[\forall x\in[-1,1]:\ |f(x)|\le 1.\] Denote $a$ the (possibly zero) coefficient of $f$ at $x^3.$ Show that there is a positive number $k$ such that \[\forall f\in\mathcal M:\ |a|\le k\] and find the least $k$ with this property.
2013 Costa Rica - Final Round, LRP2
From a set containing $6$ positive and consecutive integers they are extracted, randomly and with replacement, three numbers $a, b, c$. Determine the probability that even $a^b + c$ generates as a result .
2018 IMC, 9
Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$.
[i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]
1999 Slovenia National Olympiad, Problem 4
A pawn is put on each of $2n$ arbitrary selected cells of an $n\times n$ board ($n>1$). Prove that there are four cells that are marked with pawns and whose centers form a parallelogram.
2023 CCA Math Bonanza, L3.1
Joseph rolls a fair 6-sided dice repeatedly until he gets 3 of the same side in a row. What is the expected value of the number of times he rolls?
[i]Lightning 3.1[/i]
2005 MOP Homework, 2
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.
2013 Saudi Arabia Pre-TST, 1.2
Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.