Olympiad problems

2000 VJIMC, Problem 2

Let $f:\mathbb N\to\mathbb R$ be given by $$f(n)=n^{\frac12\tau(n)}$$for $n\in\mathbb N=\{1,2,\ldots\}$ where $\tau(n)$ is the number of divisors of $n$. Show that $f$ is an injection.

2014 NIMO Problems, 1

Tags: algebra , polynomial , induction

Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$. [i]Proposed by Evan Chen[/i]

2018 Sharygin Geometry Olympiad, 22

Tags: geometry

Six circles of unit radius lie in the plane so that the distance between the centers of any two of them is greater than $d$. What is the least value of $d$ such that there always exists a straight line which does not intersect any of the circles and separates the circles into two groups of three?

2017 ELMO Shortlist, 1

Tags: number theory , greatest common divisor

Let $a_1,a_2,\dots, a_n$ be positive integers with product $P,$ where $n$ is an odd positive integer. Prove that $$\gcd(a_1^n+P,a_2^n+P,\dots, a_n^n+P)\le 2\gcd(a_1,\dots, a_n)^n.$$ [i]Proposed by Daniel Liu[/i]

2008 IMO Shortlist, 3

Tags: geometry , circumcircle , homothety , trigonometry , quadrilateral , inversion

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

2015 Mathematical Talent Reward Programme, SAQ: P 4

Tags: algebra , number of solutions

Find all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying,$$\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)$$

1995 Miklós Schweitzer, 9

Tags: real analysis

A serpentine is a sequence of points $P_1 , ..., P_m$ in a plane, not necessarily all different, such that the distance between $P_i$ and $P_{i+1}$ is at least 1, and the segments $P_i P_{i +1}$ are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine ($P_m = P_i$ for some i < m).

2016 Regional Olympiad of Mexico Southeast, 4

Tags: geometry , quadrilateral , algebra

The diagonals of a convex quadrilateral $ABCD$ intersect in $E$. Let $S_1, S_2, S_3$ and $S_4$ the areas of the triangles $AEB, BEC, CED, DEA$ respectively. Prove that, if exists real numbers $w, x, y$ and $z$ such that $$S_1=x+y+xy, S_2=y+z+yz, S_3=w+z+wz, S_4=w+x+wx,$$ then $E$ is the midpoint of $AC$ or $E$ is the midpoint of $BD$.

Kvant 2023, M2744

Tags: geometry

A regular $100$-gon was cut into several parallelograms and two triangles. Prove that these triangles are congruent.

2020 Moldova Team Selection Test, 1

Tags: progression , arithmetic progression , geometric progression , number theory

All members of geometrical progression $(b_n)_{n\geq1}$ are members of some arithmetical progression. It is known that $b_1$ is an integer. Prove that all members of this geometrical progression are integers. (progression is infinite)

2025 CMIMC Algebra/NT, 9

Tags: algebra , number theory

Find the largest prime factor of $45^5-1.$

2010 Baltic Way, 1

Tags: system of equations , algebra

Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

1964 IMO, 4

Tags: combinatorics , ramsey theory , coloring , graph theory , extremal combinatorics

Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

2011 Pre-Preparation Course Examination, 4

Tags: combinatorics

A star $K_{1,3}$ is called a paw. suppose that $G$ is a graph without any induced paws. prove that $\chi(G)\le(\omega(G))^2$. (15 points)

2012 AIME Problems, 14

Tags: hmmt

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.

2017 All-Russian Olympiad, 1

Tags: combinatorics

In country some cities are connected by oneway flights( There are no more then one flight between two cities). City $A$ called "available" for city $B$, if there is flight from $B$ to $A$, maybe with some transfers. It is known, that for every 2 cities $P$ and $Q$ exist city $R$, such that $P$ and $Q$ are available from $R$. Prove, that exist city $A$, such that every city is available for $A$.

2011 AIME Problems, 12

Tags: probability , geometry , geometric transformation , rotation , number theory , relatively prime , complementary counting

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

1979 Canada National Olympiad, 4

Tags: calculus

A dog standing at the centre of a circular arena sees a rabbit at the wall. The rabbit runs round the wall and the dog pursues it along a unique path which is determined by running at the same speed and staying on the radial line joining the centre of the arena to the rabbit. Show that the dog overtakes the rabbit just as it reaches a point one-quarter of the way around the arena.

2003 Putnam, 2

Tags: inequalities , putnam inequalities

Let $a_1, a_2, \cdots , a_n$ and $b_1, b_2,\cdots, b_n$ be nonnegative real numbers. Show that \[(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}\]

2004 AMC 10, 8

Tags:

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $ A$, $ B$, and $ C$ start with $ 15$, $ 14$, and $ 13$ tokens, respectively. How many rounds will there be in the game? $ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 40$

2017 BMT Spring, 9

Tags: number theory , algebra

Let $a_d$ be the number of non-negative integer solutions $(a, b)$ to $a + b = d$ where $a \equiv b$ (mod $n$) for a fixed $n \in Z^+$. Consider the generating function $M(t) = a_0 + a_1t + a_2t^2 + ...$ Consider $$P(n) = \lim_{t\to 1} \left( nM(t) - \frac{1}{(1 - t)^2} \right).$$ Then $P(n)$, $n \in Z^+$ is a polynomial in $n$, so we can extend its domain to include all real numbers while having it remain a polynomial. Find $P(0)$.

2010 Contests, 4

Tags: parallelogram , incenter , perimeter , geometry

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]

2012 Uzbekistan National Olympiad, 4

Tags: inequalities , linear algebra , matrix

Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that $\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$

2024 Philippine Math Olympiad, P3

Tags: geometry , orthocenter

Given triangle $ABC$ with orthocenter $H$, the lines through $B$ and $C$ perpendicular to $AB$ and $AC$, respectively, intersect line $AH$ at $X$ and $Y$, respectively. The circle with diameter $XY$ intersects lines $BX$ and $CY$ a second time at $K$ and $L$, respectively. Prove that points $H, K$ and $L$ are collinear.

2021 Olimphíada, 2

Tags: geometry

Let $P$, $A$, $B$ and $C$ be points on a line $r$, in that order, so that $AB = BC$. Let $H$ be a point that does not belong to this line and let $S$ be the other intersection of the circles $(HPB)$ and $(HAC)$. Let $I$ be the second intersection of the circle with diameter $HB$ and $(HAC)$. Show that the points $P$, $H$, $I$ lie on the same line if and only if $HS$ is perpendicular to $r$.