Olympiad problems

2009 Ukraine Team Selection Test, 3

Let $S$ be a set consisting of $n$ elements, $F$ a set of subsets of $S$ consisting of $2^{n-1}$ subsets such that every three such subsets have a non-empty intersection. a) Show that the intersection of all subsets of $F$ is not empty. b) If you replace the number of sets from $2^{n-1}$ with $2^{n-1}-1$, will the previous answer change?

2023 Durer Math Competition Finals, 15

Tags: geometry , combinatorics , combinatorial geometry

Csongi bought a $12$-sided convex polygon-shaped pizza. The pizza has no interior point with three or more distinct diagonals passing through it. Áron wants to cut the pizza along $3$ diagonals so that exactly $6$ pieces of pizza are created. In how many ways can he do this? Two ways of slicing are different if one of them has a cut line that the other does not have.

2008 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: circumcircle , trigonometry , cyclic quadrilateral , angle bisector , geometry

In triangle $ABC$, where $AB<AC$, let $X$, $Y$, $Z$ denote the points where the incircle is tangent to $BC$, $CA$, $AB$, respectively. On the circumcircle of $ABC$, let $U$ denote the midpoint of the arc $BC$ that contains the point $A$. The line $UX$ meets the circumcircle again at the point $K$. Let $T$ denote the point of intersection of $AK$ and $YZ$. Prove that $XT$ is perpendicular to $YZ$.

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]

2023 pOMA, 1

Tags: combinatorics , permutation

Let $n$ be a positive integer. Marc has $2n$ boxes, and in particular, he has one box filled with $k$ apples for each $k=1,2,3,\ldots,2n$. Every day, Marc opens a box and eats all the apples in it. However, if he eats strictly more than $2n+1$ apples in two consecutive days, he gets stomach ache. Prove that Marc has exactly $2^n$ distinct ways of choosing the boxes so that he eats all the apples but doesn't get stomach ache.

1998 Harvard-MIT Mathematics Tournament, 10

Tags: geometry

Lukas is playing pool on a table shaped like an equilateral triangle. The pockets are at the corners of the triangle and are labeled $C$, $H$, and $T$. Each side of the table is $16$ feet long. Lukas shoots a ball from corner $C$ of the table in such a way that on the second bounce, the ball hits $2$ feet away from him along side $CH$. a. How many times will the ball bounce before hitting a pocket? b. Which pocket will the ball hit? c. How far will the ball travel before hitting the pocket?

2018 Costa Rica - Final Round, N1

Tags: consecutive , number theory

Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.

2024 LMT Fall, 13

Tags: team

$2$ identical red tokens and $2$ identical black tokens are placed on distinct cells of a $5\times5$ grid. Suppose it is impossible to color some additional cells of the grid red or black such that there exists a red path between the red tokens and a black path between the black tokens. Find the number of possible arrangements of the tokens on the grid. (A red path is a path of edge adjacent red cells, and same for a black path.)

1998 Mediterranean Mathematics Olympiad, 2

Tags: polynomial , geometry , 3d geometry , algebra

Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.

1978 AMC 12/AHSME, 11

Tags: calculus , derivative , analytic geometry , graphing lines , slope

If $r$ is positive and the line whose equation is $x + y = r$ is tangen to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals $\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{2}\qquad \textbf{(E) }2\sqrt{2}$

2006 China Second Round Olympiad, 11

Tags:

Find the number of real solutions to the equation $(x^{2006}+1)(1+x^2+x^4+\ldots +x^{2004})=2006x^{2005}$

2007 Romania National Olympiad, 3

Tags: trigonometry

Consider the triangle $ ABC$ with $ m(\angle BAC) \equal{} 90^\circ$ and $ AB < AC$.Let a point $ D$ on the side $ AC$ such that: $ m(\angle ACB) \equal{} m(\angle DBA)$.Let $ E$ be a point on the side $ BC$ such that $ DE\perp BC$.It is known that $ BD \plus{} DE \equal{} AC$. Find the measures of the angles in the triangle $ ABC$.

2023 Thailand Online MO, 10

Tags: combinatorics , combinatorial game

Let $n$ be an even positive integer. Alice and Bob play the following game. Before the start of the game, Alice chooses a set $S$ containing $m$ integers and announces it to Bob. The players then alternate turns, with Bob going first, choosing $i\in\{1,2,\dots, n\}$ that has not been chosen and setting the value of $v_i$ to either $0$ or $1$. At the end of the game, when all of $v_1,v_2,\dots,v_n$ have been set, the expression $$E=v_1\cdot 2^0 + v_2 \cdot 2^1 + \dots + v_n \cdot 2^{n-1}$$ is calculated. Determine the minimum $m$ such that Alice can always ensure that $E\in S$ regardless of how Bob plays.

2009 Thailand Mathematical Olympiad, 3

Tags: combinatorics

Teeradet is a student in a class with $19$ people. He and his classmates form clubs, so that each club must have at least one student, and each student can be in more than one club. Suppose that any two clubs differ by at least one student, and all clubs Teeradet is in have an odd number of students. What is the maximum possible number of clubs?

2012 HMNT, 8

Tags: geometry

$ABC$ is a triangle with $AB = 15$, $BC = 14$, and $CA = 13$. The altitude from $A$ to $BC$ is extended to meet the circumcircle of $ABC$ at $D$. Find $AD$.

2011 Today's Calculation Of Integral, 733

Tags: calculus , integration , limit , calculus computations

Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

2016 Saudi Arabia GMO TST, 2

Tags: inequalities , subset , combinatorics , algebra

Let $n \ge 1$ be a fixed positive integer. We consider all the sets $S$ which consist of sub-sequences of the sequence $0, 1,2, ..., n$ satisfying the following conditions: i) If $(a_i)_{i=0}^k$ belongs to $S$, then $a_0 = 0$, $a_k = n$ and $a_{i+1} - a_i \le 2$ for all $0 \le i \le k - 1$. ii) If $(a_i)_{i=0}^k$ and $(b_j)^h_{j=0}$ both belong to $S$, then there exist $0 \le i_0 \le k - 1$ and $0 \le j_0 \le h - 1$ such that $a_{i_0} = b_{j_0}$ and $a_{i_0+1} = b_{j_0+1}$. Find the maximum value of $|S|$ (among all the above-mentioned sets $S$).

Russian TST 2021, P1

Tags: number theory , representation

Do there exist infinitely many positive integers not expressible in the form \[(a+b)+\log_2(b+c)-2^{c+a},\]where $a,b,c$ are positive integers?

2010 Regional Competition For Advanced Students, 2

Tags: algebra

Solve the following in equation in $\mathbb{R}^3$: \[4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.\]

2009 Princeton University Math Competition, 3

Tags: algebra , polynomial , rational root theorem

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

2012 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: composite , number theory

Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$

2014 Canada National Olympiad, 4

Tags: circumcircle , power of a point , radical axis , geometry

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

2016 Belarus Team Selection Test, 2

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2011 Bulgaria National Olympiad, 2

Tags: relatively prime , number theory

For each natural number $a$ we denote $\tau (a)$ and $\phi (a)$ the number of natural numbers dividing $a$ and the number of natural numbers less than $a$ that are relatively prime to $a$. Find all natural numbers $n$ for which $n$ has exactly two different prime divisors and $n$ satisfies $\tau (\phi (n))=\phi (\tau (n))$.

2020 Brazil Undergrad MO, Problem 1

Tags: calculus , geometry , limit

Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of $\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?