Olympiad problems

2010 Sharygin Geometry Olympiad, 2

Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle. (Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)

2008 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , function , calculus computations , logarithm

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

1990 Romania Team Selection Test, 6

Tags: partition , subset , combinatorics

Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.

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 $EFGH$.