Found problems: 303
1978 USAMO, 5
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.
1986 USAMO, 4
Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given.
Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle.
2005 IMAR Test, 1
Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \]
[hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url].
[/hide]
1991 USAMO, 1
In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.
2011 USAMO, 6
Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.
1986 India National Olympiad, 5
If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.
2023 USAMO, 3
Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find all possible values of $k(C)$ as a function of $n$.
[i]Proposed by Holden Mui[/i]
2009 USAMO, 1
Given circles $ \omega_1$ and $ \omega_2$ intersecting at points $ X$ and $ Y$, let $ \ell_1$ be a line through the center of $ \omega_1$ intersecting $ \omega_2$ at points $ P$ and $ Q$ and let $ \ell_2$ be a line through the center of $ \omega_2$ intersecting $ \omega_1$ at points $ R$ and $ S$. Prove that if $ P, Q, R$ and $ S$ lie on a circle then the center of this circle lies on line $ XY$.
2002 USAMO, 1
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly $N$ white subsets.
2004 Germany Team Selection Test, 2
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
2016 USAMO, 3
Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY=\angle CBY$ and $\overline{BE}\perp\overline{AC}$. Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ=\angle BCZ$ and $\overline{CF}\perp\overline{AB}$.
Lines $\overleftrightarrow{I_BF}$ and $\overleftrightarrow{I_CE}$ meet at $P$. Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.
[i]Proposed by Evan Chen and Telv Cohl[/i]
2011 USAMO, 1
Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that
\[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]
2012 USAMO, 2
A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that some $108$ points are colored Red, some $108$ points are colored Green, some $108$ points are colored Blue, and the remaining $108$ points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.
1996 Moldova Team Selection Test, 12
Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.
2005 Korea - Final Round, 2
Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the
arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ ,
\[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]
2012 AIME Problems, 1
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$, $c \neq 0$ such that both $abc$ and $cba$ are divisible by 4.
2009 India IMO Training Camp, 3
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:
$ a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$.
Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.
2020 CHMMC Winter (2020-21), 9
For a positive integer $m$, let $\varphi(m)$ be the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime, and let $\sigma(m)$ be the sum of the positive divisors of $m$. Find the sum of all even positive integers $n$ such that
\[
\frac{n^5\sigma(n) - 2}{\varphi(n)}
\]
is an integer.
1997 USAMO, 2
Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.
2017 USAMO, 2
Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$, define an [i]$A$-inversion[/i] of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds:
[list]
[*]$a_i \ge w_i > w_j$
[*]$w_j > a_i \ge w_i$, or
[*]$w_i > w_j > a_i$.
[/list]
Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$, and for any positive integer $k$, the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$-inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$-inversions.
2021 USAMO, 4
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)
Given this information, find all possible values for the number of elements of $S$.
2007 Pre-Preparation Course Examination, 16
Prove that $2^{2^{n}}+2^{2^{{n-1}}}+1$ has at least $n$ distinct prime divisors.
2001 USAMO, 4
Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\angle BAC$ is acute.
2011 USAJMO, 2
Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that
\[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]
2025 USAJMO, 5
Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.