Olympiad problems

2021 Mexico National Olympiad, 4

Tags: geometry , orthocenter , euler line , circumcircle , tangent circle

Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$. [list] [*] Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point. [*] Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$. [/list] [i]Note:[/i] The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

2020 Korea - Final Round, P6

Tags: number theory , pell equation

Find all positive integers $n$ such that $6(n^4-1)$ is a square of an integer.

2013 BMT Spring, 7

Tags: number theory

Denote by $S(a,b)$ the set of integers $k$ that can be represented as $k=a\cdot m+b\cdot n$, for some non-negative integers $m$ and $n$. So, for example, $S(2,4)=\{0,2,4,6,\ldots\}$. Then, find the sum of all possible positive integer values of $x$ such that $S(18,32)$ is a subset of $S(3,x)$.

2015 BMT Spring, 9

Tags: geometry , 3d geometry , cube , geometric inequality , square

Find the side length of the largest square that can be inscribed in the unit cube.

2022 Turkey Team Selection Test, 2

Tags: functional equation , functional equation in r , algebra

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

2017 Purple Comet Problems, 8

Tags: number theory

Find the number of trailing zeros at the end of the base-$10$ representation of the integer $525^{25^2} \cdot 252^{52^5}$ .

2023 MIG, 8

Tags:

Anna is buying fruits at a grocery store. If she loses a nickel, she still has enough money to buy exactly $16$ lemons. Similarly, if she loses a quarter, she has enough money to buy exactly $14$ lemons. What is the cost of each lemon? $\textbf{(A) } \$0.05\qquad\textbf{(B) } \$0.10\qquad\textbf{(C) } \$0.15\qquad\textbf{(D) } \$0.20\qquad\textbf{(E) } \$0.25$

1983 IMO Longlists, 46

Tags: function , limit , algebra , logarithm

Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$. Suppose that \[\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.\] For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$, prove that $u_n \to +\infty$ as $n \to +\infty.$

Bangladesh Mathematical Olympiad 2020 Final, #12

Tags: number theory

$2^{2921}$ has $581$ digits and starts with a $4$. How many $2^n$'s starts with a $4$, where $0$ is the last digit?

2021/2022 Tournament of Towns, P3

Tags: geometry , incircle

The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal. Spoiler: This is one of the highlights of TT. Also SA3

2021 AMC 12/AHSME Spring, 14

Tags:

What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$ $\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$

1962 All Russian Mathematical Olympiad, 020

Tags: pentagon , minimum , maximum , geometry

Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

2022 South East Mathematical Olympiad, 6

Tags: geometry

$H$ is the orthocenter of $\triangle ABC$,the circle with center $H$ passes through $A$,and it intersects with $AC,AB$ at two other points $D,E$.The orthocenter of $\triangle ADE$ is $H'$,line $AH'$ intersects with $DE$ at point $F$.Point $P$ is inside the quadrilateral $BCDE$,so that $\triangle PDE\sim\triangle PBC$.Let point $K$ be the intersection of line $HH'$ and line $PF$.Prove that $A,H,P,K$ lie on one circle. [img]https://i.ibb.co/mcyhxRM/graph.jpg[/img]

1999 Junior Balkan Team Selection Tests - Romania, 1

Tags: algebra , polynomial

Let be a natural number $ n. $ Prove that there is a polynomial $ P\in\mathbb{Z} [X,Y] $ such that $ a+b+c=0 $ implies $$ a^{2n+1}+b^{2n+1}+c^{2n+1}=abc\left( P(a,b)+P(b,c)+P(c,a)\right) $$ [i]Dan Brânzei[/i]

1967 IMO Longlists, 28

Tags: parameterization , calculus , trigonometry , algebra , trigonometric identities

Find values of the parameter $u$ for which the expression \[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\] does not depend on $x.$

2001 District Olympiad, 2

Tags: algebra

Two numbers $(z_1,z_2)\in \mathbb{C}^*\times \mathbb{C}^*$ have the property $(P)$ if there is a real number $a\in [-2,2]$ such that $z_1^2-az_1z_2+z_2^2=0$. Prove that if $(z_1,z_2)$ have the property $(P)$, then $(z_1^n,z_2^n)$ satisfy this property, for any positive integer $n$. [i]Dorin Andrica[/i]

2003 China Team Selection Test, 2

Tags: modular arithmetic , quadratic , number theory

Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

2014 ELMO Shortlist, 1

Tags: modular arithmetic , mod 13 , number theory

Does there exist a strictly increasing infinite sequence of perfect squares $a_1, a_2, a_3, ...$ such that for all $k\in \mathbb{Z}^+$ we have that $13^k | a_k+1$? [i]Proposed by Jesse Zhang[/i]

2023 Iranian Geometry Olympiad, 3

Tags: combinatorics , combinatorial geometry , square , geometry

Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex? [i]Proposed by Josef Tkadlec - Czech Republic[/i]

2016 Fall CHMMC, 9

Tags: geometry

In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

2013 NIMO Problems, 2

Tags: geometry , perimeter , probability , analytic geometry , graphing lines , slope , geometric transformation

Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive. [i]Proposed by Isabella Grabski[/i]

2015 ASDAN Math Tournament, 15

Tags:

Let $ABCD$ be a regular tetrahedron of side length $12$. Select points $E,F,G,H$ on $AC,BC,BD,AD$, respectively, such that $AE=BF=BG=AH=3$. Compute the area of quadrilateral $EFGH$.

2024 Spain Mathematical Olympiad, 3

Tags: geometry

Let $ABC$ be a scalene triangle and $P$ be an interior point such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the internal and external bisectors of $\angle BAC$ at $Q$ and $R$, respectively. Let $S$ be the point such that $CS$ is parallel to $AQ$ and $BS$ is parallel to $AR$. Prove that $Q$, $R$ and $S$ are colinear.

2014 European Mathematical Cup, 4

Tags: function , induction , number theory

Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.

2021 Ukraine National Mathematical Olympiad, 4

Tags: geometry , circumcenter

Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$. (Fedir Yudin)