Olympiad problems

2020 Purple Comet Problems, 5

Tags: number theory

Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.

2021-IMOC, C8

Tags: tiling , combinatorics

Find all positive integers $m,n$ such that the $m \times n$ grid can be tiled with figures formed by deleting one of the corners of a $2 \times 3$ grid. [i]usjl, ST[/i]

1967 Spain Mathematical Olympiad, 1

Tags: analysis , algebra

It is known that the real function $f(t)$ is monotonic increasing in the interval $-8 \le t \le 8$, but nothing is known about what happens outside of it. In what range of values of $x$, can it be ensured that the function $y = f(2x - x^2)$ is monotonic increasing?

1992 Baltic Way, 15

Tags: search , combinatorics

Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.

2006 Tuymaada Olympiad, 4

Tags: function , algebra , induction

Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$ and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$. [i]Proposed by P. Volkmann[/i]

Kvant 2024, M2816

Tags: number theory

Find out for which natural numbers $m$ it is possible to find a natural $\ell$ such that the sum of $n+n^2+n^3+\ldots+n^\ell$ will be divisible by $m$ for any natural $n$. [i]A. Skabelin[/i]

2021 Science ON grade V, 1

Tags: number theory

Consider the prime numbers $p_1,p_2,\dots ,p_{2021}$ such that the sum $$p_1^4+p_2^4+\dots +p_{2021}^4$$ is divisible by $6060$. Prove that at least $4$ of these prime numbers are less than $2021$. $\textit{Stefan Bălăucă}$

2016 Latvia National Olympiad, 1

Tags: number theory

Given that $x$ and $y$ are positive integers such that $xy^{10}$ is perfect 33rd power of a positive integer, prove that $x^{10}y$ is also a perfect 33rd power!

2005 Belarusian National Olympiad, 6

Tags: function , number theory

$f(n+f(n))=f(n)$ for every $n \in \mathbb{N}$. a)Prove, that if $f(n)$ is finite, then $f$ is periodic. b) Give example nonperiodic function. PS. $0 \not \in \mathbb{N}$

1997 Tournament Of Towns, (562) 3

Tags: number theory , integer , perfect square

All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer. (A Kanel)

2015 Cono Sur Olympiad, 6

Tags: combinatorics , arithmetic , set

Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be [i]friends[/i] if the following conditions are true: [list] [*] They do not share any elements. [*] They both have the same number of elements. [*] The product of all elements from $A$ equals the product of all elements from $B$. [/list] Prove that there are two subsets of $S$ that are [i]friends[/i] such that each one of them contains at least $738$ elements.

2005 Tournament of Towns, 3

Tags:

Baron Münchhausen’s watch works properly, but has no markings on its face. The hour, minute and second hands have distinct lengths, and they move uniformly. The Baron claims that since none of the mutual positions of the hands is repeats twice in the period between 8:00 and 19:59, he can use his watch to tell the time during the day. Is his assertion true? [i](5 points)[/i]

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

1993 Romania Team Selection Test, 3

Tags: combinatorial geometry , combinatorics

Find all integers $n > 1$ for which there is a set $B$ of $n$ points in the plane such that for any $A \in B$ there are three points $X,Y,Z \in B$ with $AX = AY = AZ = 1$.

2022 Moscow Mathematical Olympiad, 1

Tags: number theory

There are two types of items in Alik's collection: badges and bracelets and there are more badges than bracelets. Alik noticed that if he increases the number of bracelets some (not necessarily integer) number of times without changing the number of icons, then in its collection will be $100$ items. And if, on the contrary, he increases the initial number of badges by the same number of times, leaving the same number of bracelets, then he will have $101$ items. How many badges and how many bracelets could there be in Alik's collection?

2020 Princeton University Math Competition, B2

Tags: number theory

Prove that there is a positive integer $M$ for which the following statement holds: For all prime numbers $p$, there is an integer $n$ for which $\sqrt{p} \le n \le M\sqrt{p}$ and $p \mod n \le \frac{n}{2020}$ . Note: Here, $p \mod n$ denotes the unique integer $r \in {0, 1, ..., n - 1}$ for which $n|p -r$. In other words, $p \mod n$ is the residue of $p$ upon division by $n$.

2018 Malaysia National Olympiad, B3

Tags: perfect square , number theory

Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.

2021 Lotfi Zadeh Olympiad, 1

Tags: geometry , parallel lines , cyclic quadrilateral

In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.

2022 Belarusian National Olympiad, 11.8

Tags: algebra , polynomial

A polynomial $P(x,y)$ with integer coefficients satisfies two following conditions: 1. for every integer $a$ there exists exactly one integer $y$, such that $P(a,y)=0$ 2. for every integer $b$ there exists exactly one integer $x$, such that $P(x,b)=0$ a) Prove that if the degree of $P$ is $2$, then it is divisible by either $x-y+C$ for some integer $C$, or $x+y+C$ for some integer $C$ b) Is there a polynomial $P$ that isn't divisible by any of $x-y+C$ or $x+y+C$ for integers $C$?

2002 National Chemistry Olympiad, 46

Tags:

How many orbitals contain one or more electrons in an isolated ground state iron atom (Z = 26)? $ \textbf{(A) }13 \qquad\textbf{(B) }14 \qquad\textbf{(C) } 15\qquad\textbf{(D) } 16\qquad$

2008 China Team Selection Test, 1

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

Let $ P$ be the the isogonal conjugate of $ Q$ with respect to triangle $ ABC$, and $ P,Q$ are in the interior of triangle $ ABC$. Denote by $ O_{1},O_{2},O_{3}$ the circumcenters of triangle $ PBC,PCA,PAB$, $ O'_{1},O'_{2},O'_{3}$ the circumcenters of triangle $ QBC,QCA,QAB$, $ O$ the circumcenter of triangle $ O_{1}O_{2}O_{3}$, $ O'$ the circumcenter of triangle $ O'_{1}O'_{2}O'_{3}$. Prove that $ OO'$ is parallel to $ PQ$.

2009 Purple Comet Problems, 10

Tags:

Towers grow at points along a line. All towers start with height 0 and grow at the rate of one meter per second. As soon as any two adjacent towers are each at least 1 meter tall, a new tower begins to grow at a point along the line exactly half way between those two adjacent towers. Before time 0 there are no towers, but at time 0 the first two towers begin to grow at two points along the line. Find the total of all the heights of all towers at time 10 seconds.

1985 IMO Longlists, 47

Tags: geometry

Let $F$ be the correspondence associating with every point $P = (x, y)$ the point $P' = (x', y')$ such that \[ x'= ax + b,\qquad y'= ay + 2b. \qquad (1)\] Show that if $a \neq 1$, all lines $PP'$ are concurrent. Find the equation of the set of points corresponding to $P = (1, 1)$ for $b = a^2$. Show that the composition of two mappings of type $(1)$ is of the same type.

1993 AMC 12/AHSME, 20

Tags: algebra , quadratic , complex number , quadratic formula

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true? $ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $

2019 Korea National Olympiad, 5

Tags: algebra , functional equation

Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$