Olympiad problems

2021 China Second Round A1, 2

Find a necessary and sufficient condition of $a,b,n\in\mathbb{N^*}$ such that for $S=\{a+bt\mid t=0,1,2,\cdots,n-1\}$, there exists a one-to-one mapping $f: S\to S$ such that for all $x\in S$, $\gcd(x,f(x))=1$.

1968 AMC 12/AHSME, 25

Tags:

Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is: $\textbf{(A)}\ xy \qquad\textbf{(B)}\ \frac{y}{x+y} \qquad\textbf{(C)}\ \frac{xy}{x-1} \qquad\textbf{(D)}\ \frac{x+y}{x+1} \qquad\textbf{(E)}\ \frac{x+y}{x-1}$

2015 Federal Competition For Advanced Students, 4

Tags: number theory

A [i]police emergency number[/i] is a positive integer that ends with the digits $133$ in decimal representation. Prove that every police emergency number has a prime factor larger than $7$. (In Austria, $133$ is the emergency number of the police.) (Robert Geretschläger)

2022 JHMT HS, 8

Tags: geometry , circumcircle

In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.

2018 District Olympiad, 1

Tags: number theory , fractional part

Prove that $\left\{ \frac{m}{n}\right\}+\left\{ \frac{n}{m}\right\} \ne 1$ , for any positive integers $m, n$.

2014 Baltic Way, 12

Tags: geometry , circumcircle , trapezoid , symmetry , geometric transformation , reflection , parallelogram

Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$

2007 AMC 10, 23

Tags: algebra , difference of squares , special factorizations

How many ordered pairs $ (m,n)$ of positive integers, with $ m > n$, have the property that their squares differ by $ 96$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 12$

2013 Romania National Olympiad, 4

Tags: function , inequalities , real analysis

a) Consider\[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] a differentiable and convex function .Show that $f\left( x \right)\le x$, for every $x\ge 0$, than ${f}'\left( x \right)\le 1$ ,for every $x\ge 0$ b) Determine \[f\text{:}\left[ \text{0,}\infty \right)\to \left[ \text{0,}\infty \right)\] differentiable and convex functions which have the property that $f\left( 0 \right)=0\,$, and ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$, for every $x\ge 0$

2020 OMpD, 1

Tags: algebra , factoring polynomials

Let $a, b, c$ be real numbers such that $a + b + c = 0$. Given that $a^3 + b^3 + c^3 \neq 0$, $a^2 + b^2 + c^2 \neq 0$, determine all possible values for: $$\frac{a^5 + b^5 + c^5}{(a^3 + b^3 + c^3)(a^2 + b^2 + c^2)}$$

2013 Greece Junior Math Olympiad, 3

Tags: combinatorics

Let $A=\overline{abcd}$ be a four-digit positive integer with digits $a, b, c, d$, such that $a\ge7$ and $a>b>c>d>0$. Consider the positive integer $B=\overline{dcba}$ , that comes from number $A$ by reverting the order of it's digits. Given that the number $A+B$ has all it's digits odd, find all possible values of number $A$.

2017 SDMO (High School), 4

Tags: number theory , relatively prime

For each positive integer $n$, let $\tau\left(n\right)$ be the number of positive divisors of $n$. It is well-known that if $a$ and $b$ are relatively prime positive integers then $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$. Does the converse hold? That is, if $a$ and $b$ are positive integers such that $\tau\left(ab\right)=\tau\left(a\right)\tau\left(b\right)$, then is it necessarily true that $a$ and $b$ are relatively prime? Either give a proof, or find a counter-example.

Ukraine Correspondence MO - geometry, 2013.9

Tags: geometry , cyclic quadrilateral , equal circles

Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.

2001 Moldova National Olympiad, Problem 7

Tags: set , number theory

Let $n$ be a positive integer. We denote by $S$ the sum of elements of the set $M=\{x\in\mathbb N|(n-1)^2\le x<(n+1)^2\}$. (a) Show that $S$ is divisible by $6$. (b) Find all $n\in\mathbb N$ for which $S+(1-n)(1+n)=2001$.

2004 Mexico National Olympiad, 4

Tags: combinatorics , tournament

At the end of a soccer tournament in which any pair of teams played between them exactly once, and in which there were not draws, it was observed that for any three teams $A, B$ and C, if $A$ defeated $B$ and $B$ defeated $C$, then $A$ defeated $C$. Any team calculated the difference (positive) between the number of games that it won and the number of games it lost. The sum of all these differences was $5000$. How many teams played in the tournament? Find all possible answers.

2009 Iran MO (3rd Round), 7

Tags: 3d geometry , sphere , geometry

A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge. Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces. Time allowed for this problem was 1 hour.

2008 ISI B.Stat Entrance Exam, 4

Tags: geometry

Suppose $P$ and $Q$ are the centres of two disjoint circles $C_1$ and $C_2$ respectively, such that $P$ lies outside $C_2$ and $Q$ lies outside $C_1$. Two tangents are drawn from the point $P$ to the circle $C_2$, which intersect the circle $C_1$ at point $A$ and $B$. Similarly, two tangents are drawn from the point $Q$ to the circle $C_1$, which intersect the circle $C_2$ at points $M$ and $N$. Show that $AB=MN$

2020 Malaysia IMONST 1, 18

Tags: altitude , triangle , geometry

In a triangle, the ratio of the interior angles is $1 : 5 : 6$, and the longest side has length $12$. What is the length of the altitude (height) of the triangle that is perpendicular to the longest side?

2015 Saint Petersburg Mathematical Olympiad, 4

Tags: geometry , circumcircle , parallelogram

$ABCD$ is convex quadrilateral. Circumcircle of $ABC$ intersect $AD$ and $DC$ at points $P$ and $Q$. Circumcircle of $ADC$ intersect $AB$ and $BC$ at points $S$ and $R$. Prove that if $PQRS$ is parallelogram then $ABCD$ is parallelogram

1980 AMC 12/AHSME, 24

Tags: polynomial , quadratic , rational root theorem , algebra

For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$? $\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$

2012 China Team Selection Test, 1

Tags: circumcircle , inversion , geometry

Given two circles ${\omega _1},{\omega _2}$, $S$ denotes all $\Delta ABC$ satisfies that ${\omega _1}$ is the circumcircle of $\Delta ABC$, ${\omega _2}$ is the $A$- excircle of $\Delta ABC$ , ${\omega _2}$ touches $BC,CA,AB$ at $D,E,F$. $S$ is not empty, prove that the centroid of $\Delta DEF$ is a fixed point.

2008 Ukraine Team Selection Test, 8

Tags: algebra , functional inequality

Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition \[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1 \] for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$ [i]Author: Nikolai Nikolov, Bulgaria[/i]

2016 Latvia Baltic Way TST, 11

Tags: square , combinatorics , combinatorial geometry , rectangle , geometry

Is it possible to cut a square with side $\sqrt{2015}$ into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

1994 China Team Selection Test, 3

Tags: combinatorics

For any 2 convex polygons $S$ and $T$, if all the vertices of $S$ are vertices of $T$, call $S$ a sub-polygon of $T$. [b]I. [/b]Prove that for an odd number $n \geq 5$, there exists $m$ sub-polygons of a convex $n$-gon such that they do not share any edges, and every edge and diagonal of the $n$-gon are edges of the $m$ sub-polygons. [b]II.[/b] Find the smallest possible value of $m$.

2014 AMC 10, 14

Tags: analytic geometry , graphing lines , slope , rotation , geometry

The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$? $ \textbf{(A)}\ 45\qquad\textbf{(B)}\ 48\qquad\textbf{(C)}\ 54\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 72 $

2017 China Western Mathematical Olympiad, 1

Tags: number theory

Let $p$ be a prime and $n$ be a positive integer such that $p^2$ divides $\prod_{k=1}^n (k^2+1)$. Show that $p<2n$.