Olympiad problems

2003 China Team Selection Test, 2

Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

2021 OMMock - Mexico National Olympiad Mock Exam, 6

Tags: number theory

Let $a$ and $b$ be fixed positive integers. We say that a prime $p$ is [i]fun[/i] if there exists a positive integer $n$ satisfying the following conditions: [list] [*]$p$ divides $a^{n!} + b$. [*]$p$ divides $a^{(n + 1)!} + b$. [*]$p < 2n^2 + 1$. [/list] Show that there are finitely many fun primes.

1999 Mongolian Mathematical Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

2021 Math Prize for Girls Problems, 7

Tags:

Compute the value of the infinite series \[ \sum_{k=0}^{\infty} \frac{\cos(k \pi / 4)}{2^k} \, . \]

2021 Belarusian National Olympiad, 11.7

Tags: number theory

Prove that for any positive integer $n$, there exist pairwise distinct positive integers $a,b,c$, not equal to $n$, such that $ab+n, ac+n, bc+n$ are all perfect squares.

2016 Romania National Olympiad, 3

Tags: geometry , geometric inequality , geometric inequalities

If $a, b$ and $c$ are the length of the sides of a triangle, show that $$\frac32 \le \frac{b + c}{b + c + 2a}+ \frac{a + c}{a + c + 2b}+ \frac{a + b}{a + b + 2c}\le \frac53.$$

2012 Romanian Masters In Mathematics, 4

Tags: modular arithmetic , function , number theory

Prove that there are infinitely many positive integers $n$ such that $2^{2^n+1}+1$ is divisible by $n$ but $2^n+1$ is not. [i](Russia) Valery Senderov[/i]

2014 AMC 12/AHSME, 3

Tags: symmetry

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? ${ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$

2002 All-Russian Olympiad, 2

Tags: incenter , angle bisector , geometry

Point $A$ lies on one ray and points $B,C$ lie on the other ray of an angle with the vertex at $O$ such that $B$ lies between $O$ and $C$. Let $O_1$ be the incenter of $\triangle OAB$ and $O_2$ be the center of the excircle of $\triangle OAC$ touching side $AC$. Prove that if $O_1A = O_2A$, then the triangle $ABC$ is isosceles.

1972 IMO Longlists, 23

Tags: number theory

Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\cdots a_1}$(for an arbitrary $n$) for which the following equality holds: \[\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?\]

1982 AMC 12/AHSME, 27

Tags: algebra , polynomial

Suppose $z=a+bi$ is a solution of the polynomial equation $c_4z^4+ic_3z^3+c_2z^2+ic_1z+c_0=0$, where $c_0$, $c_1$, $c_2$, $c_3$, $a$, and $b$ are real constants and $i^2=-1$. Which of the following must also be a solution? $\textbf{(A) } -a-bi\qquad \textbf{(B) } a-bi\qquad \textbf{(C) } -a+bi\qquad \textbf{(D) }b+ai \qquad \textbf{(E) } \text{none of these}$

2013 Ukraine Team Selection Test, 2

Tags: combinatorics , grid , square grid , numbers in a table

The teacher reported to Peter an odd integer $m \le 2013$ and gave the guy a homework. Petrick should star the cells in the $2013 \times 2013$ table so to make the condition true: if there is an asterisk in some cell in the table, then or in row or column containing this cell should be no more than $m$ stars (including this one). Thus in each cell of the table the guy can put at most one star. The teacher promised Peter that his assessment would be just the number of stars that the guy will be able to place. What is the greatest number will the stars be able to place in the table Petrick?

2025 AIME, 2

Tags: number theory

Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.

2024 Indonesia TST, A

Tags: algebra , function

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

2020 Regional Olympiad of Mexico Northeast, 1

Tags: floor function , algebra , recurrence relation , sequence

Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$? Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.

2009 SDMO (Middle School), 4

Tags: probability

Sally randomly chooses three different numbers from the set $\left\{1,2,\ldots,14\right\}$. What is the probability that the sum of her smallest number and her biggest number is at least $15$?

2013 India PRMO, 7

Tags: algebra , minimum

Let Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.” Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.” What is the minimum possible value of $n$ for which the above statements are true?

2014 AIME Problems, 6

Tags: probability , relatively prime , conditional probability

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\tfrac23,$ and each of the other five sides has probability $\tfrac{1}{15}.$ Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

2018 Brazil Team Selection Test, 2

Tags: algebra

Let $f(x)$ and $g(x)$ be given by $f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$ $g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$. Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.

2012 USAMO, 2

Tags: rotation , pigeonhole , probabilistic method

A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that some $108$ points are colored Red, some $108$ points are colored Green, some $108$ points are colored Blue, and the remaining $108$ points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

1996 Baltic Way, 10

Tags: relatively prime , number theory

Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including $1$ and $n$). Let $a>1$ and $n>0$ be integers such that $a^n+1$ is a prime. Prove that $d(a^n-1)\ge n$.

2007 Princeton University Math Competition, 10

Tags: geometry

$A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?

1953 AMC 12/AHSME, 8

Tags:

The value of $ x$ at the intersection of $ y\equal{}\frac{8}{x^2\plus{}4}$ and $ x\plus{}y\equal{}2$ is: $ \textbf{(A)}\ \minus{}2\plus{}\sqrt{5} \qquad\textbf{(B)}\ \minus{}2\minus{}\sqrt{5} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{none of these}$

1995 VJIMC, Problem 2

Tags: algebra , polynomial

Let $f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}$ and $f_k=f_{2n-k}$ for each $k$. Prove that $f(z)=z^ng(z+z^{-1})$, where $g$ is a polynomial of degree $n$.

1999 Cono Sur Olympiad, 5

Tags: combinatorial geometry , square , distance , combinatorics

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.