Olympiad problems

2022/2023 Tournament of Towns, P2

Tags: combinatorics , board

There is a bacterium in one of the cells of a $10 \times 10{}$ checkered board. At the first move, the bacterium shifts to a cell adjacent by side to the original one, and divides into two bacteria (both stay in the same cell). Then again, one of the bacteria on the board shifts to a cell adjacent by side and divides into two bacteria, and so on. Is it possible that after some number of such moves the number of bacteria in each cell of the board is the same? [i]Alexandr Gribalko[/i]

2008 Greece National Olympiad, 2

Tags: number theory

Find all integers $x$ and prime numbers $p$ satisfying $x^8 + 2^{2^x+2} = p$.

2020 CMIMC Geometry, 5

Tags: geometry

For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that $$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20).$$ What is $x+y$? (A [i]homothety[/i] $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$.)

2018-IMOC, C4

Tags: combinatorics , number theory

For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then $$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.

2009 Chile National Olympiad, 3

Tags: number theory , sum

Let $S = \frac{1}{a_1}+\frac{2}{a_2}+ ... +\frac{100}{a_{100}}$ where $a_1, a_2,..., a_{100}$ are positive integers. What are all the possible integer values that $S$ can take ?

2019 Kosovo Team Selection Test, 5

Tags: algebra , inequalities

$a,b,c,d$ are fixed positive real numbers. Find the maximum value of the function $f: \mathbb{R^{+}}_{0} \rightarrow \mathbb{R}$ $f(x)=\frac{a+bx}{b+cx}+\frac{b+cx}{c+dx}+\frac{c+dx}{d+ax}+\frac{d+ax}{a+bx}, x \geq 0$

1966 IMO Shortlist, 2

Tags: algebra , inequalities

Given $n$ positive real numbers $a_1, a_2, \ldots , a_n$ such that $a_1a_2 \cdots a_n = 1$, prove that \[(1 + a_1)(1 + a_2) \cdots (1 + a_n) \geq 2^n.\]

2010 South East Mathematical Olympiad, 2

Tags: algebra

Let $\mathbb{N}^*$ be the set of positive integers. Define $a_1=2$, and for $n=1, 2, \ldots,$\[ a_{n+1}=\min\{\lambda|\frac{1}{a_1}+\frac{1}{a_2}+\cdots\frac{1}{a_n}+\frac{1}{\lambda}<1,\lambda\in \mathbb{N}^*\}\] Prove that $a_{n+1}=a_n^2-a_n+1$ for $n=1,2,\ldots$.

KoMaL A Problems 2019/2020, A. 764

Tags: combinatorics , polygon

We call a diagonal of a polygon [i]nice[/i], if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$–gon with no three of its vertices being on the same line. Prove that $P$ has at least $3(n-3)/2$ nice diagonals. [i]Proposed by Bálint Hujter, Budapest and Gábor Szűcs, Szikszó[/i]

2009 Moldova Team Selection Test, 2

Tags: pigeonhole principle , geometric transformation , greatest common divisor , algebra , geometry , number theory , polynomial

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

2014 BMT Spring, 3

Tags: algebra

Suppose three boba drinks and four burgers cost $28$ dollars, while two boba drinks and six burgers cost $\$ 37.70$. If you paid for one boba drink using only pennies, nickels, dimes, and quarters, determine the least number of coins you could use.

2000 ITAMO, 6

Tags: polynomial , algebra , integer polynomial

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$. Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$. Determine all possible values of $p(1)$. (Note: $1999$ is a prime number.)

2003 Federal Competition For Advanced Students, Part 2, 1

Tags: polynomial , algebra , number theory

Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime. [hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]

2010 ELMO Shortlist, 6

Tags: inequalities

For all positive real numbers $a,b,c$, prove that \[\sqrt{\frac{a^4 + 2b^2c^2}{a^2+2bc}} + \sqrt{\frac{b^4+2c^2a^2}{b^2+2ca}} + \sqrt{\frac{c^4 + 2a^2b^2}{c^2 + 2ab}} \geq a + b + c.\] [i]In-Sung Na.[/i]

2022 Israel TST, 2

Tags: geometry , number theory , lattice points

Define a [b]ring[/b] in the plane to be the set of points at a distance of at least $r$ and at most $R$ from a specific point $O$, where $r<R$ are positive real numbers. Rings are determined by the three parameters $(O, R, r)$. The area of a ring is labeled $S$. A point in the plane for which both its coordinates are integers is called an integer point. [b]a)[/b] For each positive integer $n$, show that there exists a ring not containing any integer point, for which $S>3n$ and $R<2^{2^n}$. [b]b)[/b] Show that each ring satisfying $100\cdot R<S^2$ contains an integer point.

2000 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , 3d geometry

A number $n$ is called multiplicatively perfect if the product of all the positive divisors of $n$ is $n^2$. Determine the number of positive multiplicatively perfect numbers less than $100$.

2010 Stanford Mathematics Tournament, 4

Tags: geometry , angle bisector

Given triangle $ABC$. $D$ lies on $BC$ such that $AD$ bisects $BAC$. Given $AB=3$, $AC=9$, and $BC=8$. Find $AD$.

1955 Polish MO Finals, 5

Tags: construction , geometric inequalities , geometry

In the plane, a straight line $ m $ is given and points $ A $ and $ B $ lie on opposite sides of the straight line $ m $. Find a point $ M $ on the line $ m $ such that the difference in distances of this point from points $ A $ and $ B $ is as large as possible.

2004 Flanders Math Olympiad, 2

Tags:

Two bags contain some numbers, and the total number of numbers is prime. When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one. If the total sum of all numbers is 2004, find the number of numbers.

2014 Serbia National Math Olympiad, 2

Tags: geometry

On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$, respectively. Let $F$ ($F \neq C$) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$, and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$. If $DG=EH$, prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$. [i]Proposed by Milos Milosavljevic[/i]

1999 China National Olympiad, 1

Tags: number theory , modular arithmetic

Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and \[2m=a^{19}+b^{99}+k\cdot2^{1999}\]

2010 Cuba MO, 4

Tags: algebra , inequalities

Prove that for all positive real numbers $x, y$ holds the inequality $$x^4 + y^3 + x^2 + y + 1 > \frac92 xy.$$

2017 ASDAN Math Tournament, 1

Tags:

Clara and Nick each randomly and independently pick an integer between $0$ and $2017$, inclusive. What is the probability that the two integers they pick sum to an even number?

II Soros Olympiad 1995 - 96 (Russia), 9.6

Tags: equilateral , geometry

There is a point inside a regular triangle located at distances $5$, $6$ and $7$ from its vertices. Find the area of this regular triangle.

Oliforum Contest I 2008, 1

Tags: inequalities

Let $ a,b,c$ positive reals such that $ ab \plus{} bc \plus{} ca \equal{} 3$, show that: $ \displaystyle a^2 \plus{} b^2 \plus{} c^2 \plus{} 3 \ge \frac {a(3 \plus{} bc)^2}{(c \plus{} b)(b^2 \plus{} 3)} \plus{} \frac {b(3 \plus{} ca)^2}{(a \plus{} c)(c^2 \plus{} 3)} \plus{} \frac {c(3 \plus{} ab)^2}{(b \plus{} a)(a^2 \plus{} 3)}$ ([i]Anass BenTaleb, Ali Ben Bari High School - Taza,Morocco[/i])