Olympiad problems

2005 Federal Competition For Advanced Students, Part 1, 4

Tags: geometry , parallelogram , geometric transformation , homothety , reflection , geometry solved

We're given two congruent, equilateral triangles $ABC$ and $PQR$ with parallel sides, but one has one vertex pointing up and the other one has the vertex pointing down. One is placed above the other so that the area of intersection is a hexagon $A_1A_2A_3A_4A_5A_6$ (labelled counterclockwise). Prove that $A_1A_4$, $A_2A_5$ and $A_3A_6$ are concurrent.

2008 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , Divisibility , Extremal combinatorics , Additive combinatorics , IMO Shortlist

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

2024 LMT Fall, 9

Tags: team

Five friends named Ella, Jacob, Muztaba, Peter, and William are suspicious of their friends for having secret group chats. Call a group of three people a "secret chat" if there is a chat with just the three of them (there cannot be multiple chats with the same three people). They have the following perfectly logical conversation in this order: [list] [*] Ella: I am part of $5$ secret chats. [*] Jacob: I know all of the secret chats that Ella is in. [*] Muztaba: Peter is in all but one of my secret chats. [*] Peter: I am in a secret chat that William cannot know exists. [*] William: I share exactly two secret chats with Jacob and two secret chats with Peter. [/list] Let $E$ be the number of chats Ella is in, $J$ the number of chats Jacob is in, $M$ the number of chats Muztaba is in, $P$ the number of chats Peter is in, and $W$ the number of chats William is in. Find $10000E$ $+$ $1000J$ $+$ $100M$ $+$ $10P+W$.

2012 Mathcenter Contest + Longlist, 6

Tags: algebra , inequalities

Let $a,b,c>0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$ [i](Zhuge Liang)[/i]

CNCM Online Round 3, 2

Tags: CNCM

Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$? [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

2012 ELMO Shortlist, 4

Tags: factorial , number theory proposed , number theory

Do there exist positive integers $b,n>1$ such that when $n$ is expressed in base $b$, there are more than $n$ distinct permutations of its digits? For example, when $b=4$ and $n=18$, $18 = 102_4$, but $102$ only has $6$ digit arrangements. (Leading zeros are allowed in the permutations.) [i]Lewis Chen.[/i]

2023 Irish Math Olympiad, P6

Tags: number theory

A positive integer is [i]totally square[/i] is the sum of its digits (written in base $10$) is a square number. For example, $13$ is totally square because $1 + 3 = 2^2$, but $16$ is not totally square. Show that there are infinitely many positive integers that are not the sum of two totally square integers.

2000 Taiwan National Olympiad, 2

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

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

2007 Pre-Preparation Course Examination, 10

Tags: number theory unsolved , number theory

Let $a >1$ be a positive integer. Prove that the set $\{a^2+a-1,a^3+a-1,\cdots\}$ have a subset $S$ with infinite members and for any two members of $S$ like $x,y$ we have $\gcd(x,y)=1$. Then prove that the set of primes has infinite members.

2005 Georgia Team Selection Test, 9

Tags: induction , algebra proposed , algebra

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

Today's calculation of integrals, 867

Tags: calculus , integration , quadratics , function , derivative , algebra , linear equation

Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

2018 Bulgaria EGMO TST, 1

Tags: Tournament , combinatorics , results

In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a loss is worth $0$ points. Determine the smallest positive integer $n$ for which it is possible that after the $n$-th match all teams have a different number of points and each team has a non-zero number of points.

1984 All Soviet Union Mathematical Olympiad, 383

Tags: integer roots , trinomial , polynomial , game , combinatorics

The teacher wrote on a blackboard: $$x^2 + 10x + 20$$ Then all the pupils in the class came up in turn and either decreased or increased by $1$ either the free coefficient or the coefficient at $x$, but not both. Finally they have obtained: $$x^2 + 20x + 10$$ Is it true that some time during the process there was written the square polynomial with the integer roots?

2009 Iran MO (2nd Round), 2

Tags: combinatorics proposed , combinatorics

In some of the $ 1\times1 $ squares of a square garden $ 50\times50 $ we've grown apple, pomegranate and peach trees (At most one tree in each square). We call a $ 1\times1 $ square a [i]room[/i] and call two rooms [i]neighbor[/i] if they have one common side. We know that a pomegranate tree has at least one apple neighbor room and a peach tree has at least one apple neighbor room and one pomegranate neighbor room. We also know that an empty room (a room in which there’s no trees) has at least one apple neighbor room and one pomegranate neighbor room and one peach neighbor room. Prove that the number of empty rooms is not greater than $ 1000. $

2015 India IMO Training Camp, 3

Tags: trigonometry , inequalities

Prove that for any triangle $ABC$, the inequality $\displaystyle\sum_{\text{cyclic}}\cos A\le\sum_{\text{cyclic}}\sin (A/2)$ holds.

2009 Germany Team Selection Test, 2

Tags: function , algebra , Functional inequality , IMO Shortlist

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

2017 Romania National Olympiad, 3

Tags: geometry , parallelogram , square

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

2011 Harvard-MIT Mathematics Tournament, 4

Tags: HMMT , function

For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.

2021 Purple Comet Problems, 27

Tags: Purple Comet

Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.

2025 Harvard-MIT Mathematics Tournament, 9

Tags: team

Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$ $\quad \bullet \ f(x+p)=f(x)\text{ and}$ $\quad \bullet \ p \text{ divides } f(x+f(x))-x.$

2024 Azerbaijan IZhO TST, 4

Tags: number theory , TST , AZE IzHO TST

Take a sequence $(a_n)_{n=1}^\infty$ such that $a_1=3$ $a_n=a_1a_2a_3...a_{n-1}-1$ [b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence. [b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.

2022 Vietnam National Olympiad, 2

Tags: vmo , combinatorics , probability

We are given 4 similar dices. Denote $x_i (1\le x_i \le 6)$ be the number of dots on a face appearing on the $i$-th dice $1\le i \le 4$ a) Find the numbers of $(x_1,x_2,x_3,x_4)$ b) Find the probability that there is a number $x_j$ such that $x_j$ is equal to the sum of the other 3 numbers c) Find the probability that we can divide $x_1,x_2,x_3,x_4$ into 2 groups has the same sum

2000 India National Olympiad, 3

Tags: algebra solved , algebra

If $a,b,c,x$ are real numbers such that $abc \not= 0$ and \[ \frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c}, \] then prove that $a = b = c$.

1989 National High School Mathematics League, 6

Tags:

Set $M=\{u|u=12m+8n+4l,m,n,l\in\mathbb{Z}\},N=\{u|u=20p+16q+12x,p,q,x\in\mathbb{Z}\}$. Then $\text{(A)}M=N\qquad\text{(B)}M\not\subset N,N\not\subset M\qquad\text{(C)}M\subset N\qquad\text{(D)}N\subset M$

2023 Assara - South Russian Girl's MO, 4

Tags: combinatorics , Coloring

In a $50 \times 50$ checkered square, each cell is painted in one of $100$ given colors so that all colors are present and it is impossible to cut a single-color domino from the square (i.e. a $1 \times 2$ rectangle). Galiia wants to recolor all the cells of one of the colors into another color (out of the given $100$ colors) so that this condition is preserved (i.e., it is still impossible to cut out a domino of the same color). Is it true that Galiia will definitely be able to do this?