Olympiad problems

1999 Bosnia and Herzegovina Team Selection Test, 3

Let $f : [0,1] \rightarrow \mathbb{R}$ be injective function such that $f(0)+f(1)=1$. Prove that exists $x_1$, $x_2 \in [0,1]$, $x_1 \neq x_2$ such that $2f(x_1)<f(x_2)+\frac{1}{2}$. After that state at least one generalization of this result

2007 Macedonia National Olympiad, 1

Tags: inequalities

Let $a, b, c$ be positive real numbers. Prove that \[1+\frac{3}{ab+bc+ca}\geq\frac{6}{a+b+c}.\]

2007 Grigore Moisil Intercounty, 3

Tags: function , injectivity , algebra , functional equation

Find the natural numbers $ a $ that have the property that there exists a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ f(f(n))=a+n, $ for any natural number $ n, $ and the function $ g:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $ g(n)=f(n)-n $ is injective.

2015 USA TSTST, 5

Tags: number theory , relatively prime , function , analytic number theory

Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$. [i]Proposed by Iurie Boreico[/i]

2024 Chile National Olympiad., 6

Tags: number theory

Let $ 133\ldots 33 $ be a number with $ k \geq 2 $ digits, which we assume is prime. Prove that $ k(k + 2) $ is a multiple of 24. (For example, 133...33 is a prime number when $ k = 16$

2013 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , geometry , incenter , circumcircle , geometric transformation , reflection , analytic geometry

Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.

2022 China Northern MO, 1

Tags: isosceles right triangle , equal angles , geometry

As shown in the figure, given $\vartriangle ABC$ with $AB \perp AC$, $AB=BC$, $D$ is the midpoint of the side $AB$, $DF\perp DE$, $DE=DF$ and $BE \perp EC$. Prove that $\angle AFD= \angle CEF$. [img]https://cdn.artofproblemsolving.com/attachments/9/2/f16a8c8c463874f3ccb333d91cdef913c34189.png[/img]

2006 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , trapezoid

Suppose $ABCD$ is an isosceles trapezoid in which $\overline{AB}\parallel\overline{CD}$. Two mutually externally tangent circles $\omega_1$ and $\omega_2$ are inscribed in $ABCD$ such that $\omega_1$ is tangent to $\overline{AB}$,$\overline{BC}$, and $\overline{CD}$ while $\omega_2$ is tangent to $\overline{AB}$, $\overline{DA}$, and $\overline{CD}$. Given that $AB=1$, $CD=6$, compute the radius of either circle.

2018 Hanoi Open Mathematics Competitions, 9

Tags: combinatorics , cube

How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point.

2006 Mexico National Olympiad, 2

Tags: geometry , similar triangles

Let $ABC$ be a right triangle with a right angle at $A$, such that $AB < AC$. Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$. Let $E$ be the intersection of the parallel to $AC$ that passes through $M$, with the perpendicular on $BD$ passing through $B$. Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$.

Geometry Mathley 2011-12, 6.3

Tags: circles , tangent circle , geometry

Let $AB$ be an arbitrary chord of the circle $(O)$. Two circles $(X)$ and $(Y )$ are on the same side of the chord $AB$ such that they are both internally tangent to $(O)$ and they are tangent to $AB$ at $C,D$ respectively, $C$ is between $A$ and $D$. Let $H$ be the intersection of $XY$ and $AB, M$ the midpoint of arc $AB$ not containing $X$ and $Y$ . Let $HM$ meet $(O)$ again at $I$. Let $IX, IY$ intersect $AB$ again at $K, J$. Prove that the circumcircle of triangle $IKJ$ is tangent to $(O)$. Nguyễn Văn Linh

Croatia MO (HMO) - geometry, 2011.7

Tags: circumcircle , perpendicular bisector , projective geometry , power of a point , radical axis , geometry

Let $K$ and $L$ be the points on the semicircle with diameter $AB$. Denote intersection of $AK$ and $AL$ as $T$ and let $N$ be the point such that $N$ is on segment $AB$ and line $TN$ is perpendicular to $AB$. If $U$ is the intersection of perpendicular bisector of $AB$ an $KL$ and $V$ is a point on $KL$ such that angles $UAV$ and $UBV$ are equal. Prove that $NV$ is perpendicular to $KL$.

2016 Purple Comet Problems, 7

Tags:

Positive integers m and n are both greater than 50, have a least common multiple equal to 480, and have a greatest common divisor equal to 12. Find m + n.

2005 Greece Team Selection Test, 1

Tags: algebra , polynomial , root

The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.

2010 Contests, 3

Tags: induction , combinatorics

Determine all possible values of positive integer $n$, such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.

2014 Putnam, 2

Tags: linear algebra , matrix , induction , putnam matrices

Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

2012 Tuymaada Olympiad, 3

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

Point $P$ is taken in the interior of the triangle $ABC$, so that \[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\] Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$. [i]Proposed by S. Berlov[/i]

2014 ELMO Shortlist, 7

Tags: number theory

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

2022 Girls in Mathematics Tournament, 3

Tags: combinatorics

There are $n$ cards. Max and Lewis play, alternately, the following game Max starts the game, he removes exactly $1$ card, in each round the current player can remove any quantity of cards, from $1$ card to $t+1$ cards, which $t$ is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of $n$ such that Max has the winning strategy.

MBMT Team Rounds, 2020.3

Tags:

Square $ABCD$ has a side length of 1. Point $E$ lies on the interior of $ABCD$, and is on the line $\overleftrightarrow{AC}$ such that the length of $\overline{AE}$ is 1. Find the shortest distance from point $E$ to a side of square $ABCD$. [i]Proposed by Chris Tong[/i]

2012 NIMO Problems, 2

Tags: inequalities

Compute the number of positive integers $n$ satisfying the inequalities \[ 2^{n-1} < 5^{n-3} < 3^n. \][i]Proposed by Isabella Grabski[/i]

2017 ELMO Shortlist, 2

Tags: inequalities , algebra

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

2019 India PRMO, 25

Tags: geometry

Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB,AC$ and $BD$ are integers and $AB-BD$ $=$ $3$, find $AC$.

2024 CCA Math Bonanza, T5

Tags:

Find the number of permutations of the numbers $1,1,2,2,3,3,4,4$ such that no two consecutive numbers are equal. [i]Team #5[/i]

1994 Chile National Olympiad, 1

Tags: algebra , combinatorics

A railway line is divided into ten sections by stations $E_1, E_2,..., E_{11}$. The distance between the first and the last station is $56$ km. A trip through two consecutive stations never exceeds $ 12$ km, and a trip through three consecutive stations is at least $17$ Km. Calculate the distance between $E_2$ and $E_7$.