Olympiad problems

2022 China Northern MO, 1

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 $E_7$.

2017 VTRMC, 6

Tags:

Let $ f ( x ) \in \mathbb { Z } [ x ] $ be a polynomial with integer coefficients such that $ f ( 1 ) = - 1 , f ( 4 ) = 2 $ and $f ( 8 ) = 34 $. Suppose $n\in\mathbb{Z}$ is an integer such that $ f ( n ) = n ^ { 2 } - 4 n - 18 $. Determine all possible values for $n$.

1954 Polish MO Finals, 6

Tags: geometry

Inside a hoop of radius $ 2r $ a disk of radius $ r $ rolls on the hoop without slipping. What line is traced by a point arbitrarily chosen on the edge of the disk?

2010 Indonesia TST, 4

Tags: combinatorics

$300$ parliament members are divided into $3$ chambers, each chamber consists of $100$ members. For every $2$ members, they either know each other or are strangers to each other.Show that no matter how they are divided into these $3$ chambers, it is always possible to choose $2$ members, each from different chamber such that there exist $17$ members from the third chamber so that all of them knows these two members, or all of them are strangers to these two members.

2025 Romania EGMO TST, P4

Tags: binomial coefficient , number theory , least common multiple

How does one show $$\text{lcm}\left(\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\right)=\frac{\text{lcm}(1,2,\ldots,n+1)}{n+1}$$

2020 Peru Cono Sur TST., P6

Tags: number theory

Let $a_1, a_2, a_3, \ldots$ a sequence of positive integers that satisfy the following conditions: $$a_1=1, a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor}, \forall n\ge 1$$ Prove that for every positive integer $k$ there exists a term $a_i$ that is divisible by $k$

2005 Romania National Olympiad, 1

Tags: ratio , quadratic , analytic geometry , vector , gauss , geometry

Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$. [i]Virgil Nicula[/i]