Olympiad problems

1975 USAMO, 4

Tags:

Two given circles intersect in two points $ P$ and $ Q$. Show how to construct a segment $ AB$ passing through $ P$ and terminating on the circles such that $ AP \cdot PB$ is a maximum.

1979 IMO Longlists, 55

Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.

1996 Tournament Of Towns, (484) 2

Tags: number theory , perfect square

Does there exist an integer n such that all three numbers (a) $n - 96$, $n$ and $n + 96$ (b) $n - 1996$, $n$ and $n + 1996$ are positive prime numbers? (V Senderov)

2011 Mongolia Team Selection Test, 3

Tags: function , graph theory , combinatorics

Let $n$ and $d$ be positive integers satisfying $d<\dfrac{n}{2}$. There are $n$ boys and $n$ girls in a school. Each boy has at most $d$ girlfriends and each girl has at most $d$ boyfriends. Prove that one can introduce some of them to make each boy have exactly $2d$ girlfriends and each girl have exactly $2d$ boyfriends. (I think we assume if a girl has a boyfriend, she is his girlfriend as well and vice versa) (proposed by B. Batbaysgalan, folklore).

2023 China Team Selection Test, P22

Tags: algebra , functional equation , function

Find all functions $f:\mathbb {Z}\to\mathbb Z$, satisfy that for any integer ${a}$, ${b}$, ${c}$, $$2f(a^2+b^2+c^2)-2f(ab+bc+ca)=f(a-b)^2+f(b-c)^2+f(c-a)^2$$

2025 Bulgarian Winter Tournament, 12.1

Tags: inequalities , trigonometry , function

Let $a,b,c$ be positive real numbers with $a+b>c$. Prove that $ax + \sin(bx) + \cos(cx) > 1$ for all $x\in \left(0, \frac{\pi}{a+b+c}\right)$.

2024 Korea - Final Round, P6

Tags: number theory

Prove that there exists a positive integer $K$ that satisfies the following condition. Condition: For any prime $p > K$, the number of positive integers $a \le p$ that $p^2 \mid a^{p-1} - 1$ is less than $\frac{p}{2^{2024}}$

1997 Italy TST, 4

Tags: combinatorics

There are $n$ pawns on $n$ distinct squares of a $19\times 19$ chessboard. In each move, all the pawns are simultaneously moved to a neighboring square (horizontally or vertically) so that no two are moved onto the same square. No pawn can be moved along the same line in two successive moves. What is largest number of pawns can a player place on the board (being able to arrange them freely) so as to be able to continue the game indefinitely?

2018 HMIC, 5

Tags: college , combinatorics

Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine equation , diophantine

Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

2004 Greece JBMO TST, 4

Tags: algebra , inequalities

Let $a,b$ be positive real numbers such that $b^3+b\le a-a^3$. Prove that: i) $b<a<1$ ii) $a^2+b^2<1$

2013 Indonesia MO, 5

Tags: polynomial , quadratic , algebra

Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that: - $P(x) = P_1(x) + P_2(x) + P_3(x)$ - $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root) - The roots of $P_1, P_2, P_3$ are different

2024 AMC 12/AHSME, 12

Tags: geometric sequence

The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$ $\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$

2020 Brazil Cono Sur TST, 2

Tags: number theory

A number n is called charming when $ 4k^2 + n $ is a prime number for every $ 0 \leq k <n $ integer, find all charming numbers.

2022 Caucasus Mathematical Olympiad, 6

Tags: combinatorics

16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it happens that $k$ teams have non-negative balance of wins (for example, the team that won in the first round with a score of 4-2 and lost in the second with a score of 4-3 fits the condition: it has $4+3=7$ wins and $2+4=6$ losses). Find the least possible $k$.

2009 Bundeswettbewerb Mathematik, 2

Tags: prime divisor , number theory , quadratic trinomial , trinomial , prime , divisor

Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

2019 Iran MO (3rd Round), 1

Tags: geometry , algebra

Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that: $\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $ Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.

1974 IMO Longlists, 12

Tags: parallelogram , geometry

A circle $K$ with radius $r$, a point $D$ on $K$, and a convex angle with vertex $S$ and rays $a$ and $b$ are given in the plane. Construct a parallelogram $ABCD$ such that $A$ and $B$ lie on $a$ and $b$ respectively, $SA+SB=r$, and $C$ lies on $K$.

2013 National Olympiad First Round, 27

Tags: algebra , polynomial , inequalities , trigonometry

For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 0 $

2014 Rioplatense Mathematical Olympiad, Level 3, 4

Tags: number theory , sets of integers , divisor

A pair (a,b) of positive integers is [i]Rioplatense [/i]if it is true that $b + k$ is a multiple of $a + k$ for all $k \in\{ 0 , 1 , 2 , 3 , 4 \}$. Prove that there is an infinite set $A$ of positive integers such that for any two elements $a$ and $b$ of $A$, with $a < b$, the pair $(a,b)$ is [i]Rioplatense[/i].

2015 Romanian Master of Mathematics, 1

Tags: number theory , relatively prime

Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

2014 Saudi Arabia BMO TST, 4

Tags: inequalities , triangle inequality , geometry

Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )

2022 Purple Comet Problems, 25

Tags: geometry

Let $ABCD$ be a parallelogram with diagonal $AC = 10$ such that the distance from $A$ to line $CD$ is $6$ and the distance from $A$ to line $BC$ is $7$. There are two non-congruent configurations of $ABCD$ that satisfy these conditions. The sum of the areas of these two parallelograms is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2017-2018 SDPC, 3

Tags: geometry , perpendicular bisector

Let $n > 2$ be a fixed positive integer. For a set $S$ of $n$ points in the plane, let $P(S)$ be the set of perpendicular bisectors of pairs of distinct points in $S$. Call set $S$ [i]complete[/i] if no two (distinct) pairs of points share the same perpendicular bisector, and every pair of lines in $P(S)$ intersects. Let $f(S)$ be the number of distinct intersection points of pairs of lines in $P(S)$. (a) Find all complete sets $S$ such that $f(S) = 1$. (b) Let $S$ be a complete set with $n$ points. Show that if $f(S)>1$, then $f(S) \geq n$.

MOAA Gunga Bowls, 2021.24

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Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number? [i]Proposed by Andy Xu[/i]