Olympiad problems

1979 Putnam, B6

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For $k=1,2 \dots, n$ let $z_k=x_k+iy_k,$ where the $x_k$ and $y_k$ are real and $i=\sqrt{-1}$. Let $r$ bet the absolute value of the real part of $$\pm \sqrt{z_1^2+z_2^2+\dots z_n^2}.$$ Prove that $r\leq |x_1|+|x_2|+ \dots +|x_n|.$

2016 Harvard-MIT Mathematics Tournament, 8

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Let $X$ be the collection of all functions $f: \{0,1,\dots, 2016\} \rightarrow \{0,1,\dots, 2016\}$. Compute the number of functions $f \in X$ such that \[ \max_{g \in X} \left( \min_{0 \le i \le 2016} \big( \max (f(i), g(i)) \big) - \max_{0 \le i \le 2016} \big( \min (f(i),g(i)) \big) \right) = 2015. \]

2024-25 IOQM India, 23

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Consider the fourteen numbers, $1^4,2^4,...,14^4$. The smallest natural numebr $n$ such that they leave distinct remainders when divided by $n$ is:

PEN S Problems, 4

If $x$ is a real number such that $x^2 -x$ is an integer, and for some $n \ge 3$, $x^n -x$ is also an integer, prove that $x$ is an integer.

JBMO Geometry Collection, 2017

Tags: geometry

Let $ABC $ be an acute triangle such that $AB\neq AC$ ,with circumcircle $ \Gamma$ and circumcenter $O$. Let $M$ be the midpoint of $BC$ and $D$ be a point on $ \Gamma$ such that $AD \perp BC$. let $T$ be a point such that $BDCT$ is a parallelogram and $Q$ a point on the same side of $BC$ as $A$ such that $\angle{BQM}=\angle{BCA}$ and $\angle{CQM}=\angle{CBA}$. Let the line $AO$ intersect $ \Gamma$ at $E$ $(E\neq A)$ and let the circumcircle of $\triangle ETQ$ intersect $ \Gamma$ at point $X\neq E$. Prove that the point $A,M$ and $X$ are collinear.

2000 Regional Competition For Advanced Students, 4

Tags: number theory , algebra , rational , sequence , recurrence relation

We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$. (a) Determine the terms of the sequence for $u_1 = 1$. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.

2011 All-Russian Olympiad Regional Round, 9.8

Tags: geometry , combinatorics , rectangle

Straight rod of 2 meter length is cut into $N$ sticks. The length of each piece is an integer number of centimeters. For which smallest $N$ can one guarantee that it is possible to form the contour of some rectangle, while using all sticks and not breaking them further? (Author: A. Magazinov)

2023 Poland - Second Round, 2

Tags: grid , combinatorics

Let $n \geq 2$ be an integer. A lead soldier is moving across the unit squares of a $n \times n$ grid, starting from the corner square. Before each move to the neighboring square, the lead soldier can (but doesn't need to) turn left or right. Determine the smallest number of turns, which it needs to do, to visit every square of the grid at least once. At the beginning the soldier's back is faced at the edge of the grid.

Novosibirsk Oral Geo Oly VIII, 2019.4

Tags: equal angles , geometry

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

2025 Malaysian IMO Team Selection Test, 3

Tags: inequalities , am-gm

Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$ holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$. [i]Proposed by Ivan Chan Kai Chin[/i]

2009 IberoAmerican, 3

Tags: symmetry , circumcircle , parallelogram , geometry

Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL \equal{} \angle XDC \equal{} 30^\circ$ and $ CX \equal{} O_1O_2$. [i] Author: Arnoldo Aguilar (El Salvador)[/i]

2020 Peru IMO TST, 1

Tags: number theory , perfect power , prime number

Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2003 Singapore MO Open, 1

Tags: combinatorics , word , alphabet

A sequence $(a_1,a_2,...,a_{675})$ is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence $a$ is never followed by $b$ and $c$ is never followed by $d$. Show that there are integers $m$ and $n$ with $1 \le m < n \le 674$ such that $a_m = a_n$ and $a_{m+1} = a_{n+1}$·

1989 Mexico National Olympiad, 3

Tags: sum of digits , number theory

Prove that there is no $1989$-digit natural number at least three of whose digits are equal to $5$ and such that the product of its digits equals their sum.

2007 China Northern MO, 4

Tags: inequalities , area of a triangle , algebra , geometry , inradius , heron's formula , triangle inequality

The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

2009 Ukraine Team Selection Test, 6

Tags: number theory , set , modulo , remainder

Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.

LMT Speed Rounds, 2011.11

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Let $ABCD$ be a convex quadrilateral with $AB=AD, m\angle A = 40^{\circ}, m\angle C = 130^{\circ},$ and $m\angle ADC - m\angle ABC = 20^{\circ}.$ Find the measure of the non-reflex angle $\angle CDB$ in degrees.

1967 German National Olympiad, 4

Tags: geometry , polygon , counting , obtuse triangle

Given a regular $n$-gon $A_{1}A_{2}...A_{n}$ (with $n\geq 3$) in a plane. How many triangles of the kind $A_{i}A_{j}A_{k}$ are obtuse ?

2012 Iran Team Selection Test, 2

Tags: polynomial , algebra , pigeonhole principle

Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers? [i]Proposed by Morteza Saghafian[/i]

2018-2019 SDML (High School), 3

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How many three-digit positive integers $x$ are there with the property that $x$ and $2x$ have only even digits? (One such number is $x = 220$, since $2x = 440$ and each of $x$ and $2x$ has only even digits.) $ \mathrm{(A) \ } 16 \qquad \mathrm{(B) \ } 18 \qquad \mathrm {(C) \ } 64 \qquad \mathrm{(D) \ } 100 \qquad \mathrm{(E) \ } 125$

2000 Canada National Olympiad, 3

Tags: combinatorics

Let $A = (a_1, a_2, \cdots ,a_{2000})$ be a sequence of integers each lying in the interval $[-1000,1000]$. Suppose that the entries in A sum to $1$. Show that some nonempty subsequence of $A$ sums to zero.

2015 USAMTS Problems, 5

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Find all positive integers $n$ that have distinct positive divisors $d_1, d_2, \dots, d_k$, where $k>1$, that are in arithmetic progression and $$n=d_1+d_2+\cdots+d_k.$$ Note that $d_1, d_2, \dots, d_k$ do not have to be all the divisors of $n$.

1984 IMO Longlists, 45

Tags: algebra

Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$.

1997 All-Russian Olympiad, 1

Tags: modular arithmetic , algebra , quadratic , polynomial

Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots? [i]N. Agakhanov[/i]

2023 VIASM Summer Challenge, Problem 1

Tags: algebra , inequalities

Find the largest positive real number $k$ such that the inequality$$a^3+b^3+c^3-3\ge k(3-ab-bc-ca)$$holds for all positive real triples $(a;b;c)$ satisfying $a+b+c=3.$