Olympiad problems

2013 Philippine MO, 1

Tags:

1. Determine, with proof, the least positive integer $n$ for which there exist $n$ distinct positive integers, $\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}$

2023 China National Olympiad, 2

Tags: geometry

Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$. Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$. Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$.

2002 Canada National Olympiad, 5

Tags: function , modular arithmetic , functional equation , algebra

Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that \[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \] for all $x$ and $y$ in $\mathbb N$.

2013 Purple Comet Problems, 11

Tags: percent

After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than her walking all the way to and from the store without her bicycle.

1985 IMO Longlists, 31

Tags: ellipse , concurrency , conic , geometry

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

2007 Stanford Mathematics Tournament, 7

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Find the minimum value of $xy+x+y+\frac{1}{xy}+\frac{1}{x}+\frac{1}{y}$ for $x, y>0$ real.

1992 Tournament Of Towns, (349) 1

Tags: geometry , 3d geometry , combinatorics , combinatorial geometry

We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.) (A Spivak)

2004 Baltic Way, 17

Tags: rectangle , geometry , inequalities

Consider a rectangle with sidelengths 3 and 4, pick an arbitrary inner point on each side of this rectangle. Let $x, y, z$ and $u$ denote the side lengths of the quadrilateral spanned by these four points. Prove that $25 \leq x^2+y^2+z^2+u^2 \leq 50$.

1998 Bulgaria National Olympiad, 3

Tags: trigonometry , complex number , geometry

On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.

2018 Polish MO Finals, 5

Tags: radical axis , projective geometry , geometry

An acute triangle $ABC$ in which $AB<AC$ is given. Points $E$ and $F$ are feet of its heights from $B$ and $C$, respectively. The line tangent in point $A$ to the circle escribed on $ABC$ crosses $BC$ at $P$. The line parallel to $BC$ that goes through point $A$ crosses $EF$ at $Q$. Prove $PQ$ is perpendicular to the median from $A$ of triangle $ABC$.

2007 AMC 12/AHSME, 25

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Call a set of integers [i]spacy[/i] if it contains no more than one out of any three consecutive integers. How many subsets of $ \{1,2,3,\ldots,12\},$ including the empty set, are spacy? $ \textbf{(A)}\ 121 \qquad \textbf{(B)}\ 123 \qquad \textbf{(C)}\ 125 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 129$

2014 Contests, 2

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Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?

2009 Costa Rica - Final Round, 2

Tags: induction , number theory

Prove that for that for every positive integer $ n$, the smallest integer that is greater than $ (\sqrt {3} \plus{} 1)^{2n}$ is divisible by $ 2^{n \plus{} 1}$.

2013 Argentina National Olympiad, 6

Tags: number theory

A positive integer $n$ is called [i]pretty[/i] if there exists two divisors $d_1,d_2$ of $n$ $(1\leq d_1,d_2\leq n)$ such that $d_2-d_1=d$ for each divisor $d$ of $n$ (where $1<d<n$). Find the smallest pretty number larger than $401$ that is a multiple of $401$.

2009 Romania National Olympiad, 3

Tags: function , find all functions , algebra

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$ for all real numbers $ x,y. $

1966 IMO Longlists, 24

Tags: graph theory , vertex degree , combinatorics

There are $n\geq 2$ people at a meeting. Show that there exist two people at the meeting who have the same number of friends among the persons at the meeting. (It is assumed that if $A$ is a friend of $B,$ then $B$ is a friend of $A;$ moreover, nobody is his own friend.)

2004 Greece Junior Math Olympiad, 1

Tags: number theory

The numbers $203$ and $298$ divided with the positive integer $x$ give both remainder $13$. Which are the possible values of $x$ ?

2003 Junior Balkan Team Selection Tests - Moldova, 4

Tags: number theory

Let $m$ and $n$ be arbitrary positive integers, and $a, b, c$ be different natural numbers of the form $2^m.5^n$. Determine the number of all equations of the form $ax^2-2bx+c=0$ if it is known that each equation has only one real solution.

2025 Vietnam Team Selection Test, 3

Tags: combinatorics , graph

In a summer camp about Applied Maths, there are $8m+1$ boys (with $m > 5$) and some girls. Every girl is friend with exactly $3$ boys and for any $2$ boys, there is exactly $1$ girl who is their common friend. Let $n$ be the greatest number of girls that can be chosen from the camp to form a group such that every boy is friend with at most $1$ girl in the group. Prove that $n \geq 2m+1$.

1956 AMC 12/AHSME, 2

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Mr. Jones sold two pipes at $ \$ 1.20$ each. Based on the cost, his profit one was $ 20 \%$ and his loss on the other was $ 20 \%$. On the sale of the pipes, he: $ \textbf{(A)}\ \text{broke even} \qquad\textbf{(B)}\ \text{lost } 4\text{ cents} \qquad\textbf{(C)}\ \text{gained } 4\text{ cents} \qquad\textbf{(D)}\ \text{lost } 10 \text{ cents} \qquad\textbf{(E)}\ \text{gained } 10 \text{ cents}$

2003 Vietnam National Olympiad, 1

Tags: function , trigonometry , calculus , algebra

Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$

2021 Ukraine National Mathematical Olympiad, 6

Tags: geometry , symmedian , bisects segment

The altitudes $AA_1, BB_1$ and $CC_1$ were drawn in the triangle $ABC$. Point $K$ is a projection of point $B$ on $A_1C_1$. Prove that the symmmedian $\vartriangle ABC$ from the vertex $B$ divides the segment $B_1K$ in half. (Anton Trygub)

2012 AIME Problems, 1

Tags: modular arithmetic

Find the number of ordered pairs of positive integer solutions $(m,n)$ to the equation $20m+12n=2012.$

2024 Kurschak Competition, 3

Tags: number theory

Let $p$ be a prime and $H\subseteq \{0,1,\ldots,p-1\}$ a nonempty set. Suppose that for each element $a\in H$ there exist elements $b$, $c\in H\setminus \{a\}$ such that $b+ c-2a$ is divisible by $p$. Prove that $p<4^k$, where $k$ denotes the cardinality of $H$.

2021 Thailand TSTST, 1

Tags: prime factorization , divisor , number theory

For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$