Olympiad problems

2011 Iran Team Selection Test, 12

Tags: function , pigeonhole principle , induction , modular arithmetic , number theory

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

1989 Bulgaria National Olympiad, Problem 3

Tags: polynomial , algebra , irreducibility

Let $p$ be a real number and $f(x)=x^p-x+p$. Prove that: (a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

2018 India Regional Mathematical Olympiad, 1

Tags: geometry , circumcircle , angle bisector

Let $ABC$ be an acute angled triangle and let $D$ be an interior point of the segment $BC$. Let the circumcircle of $ACD$ intersect $AB$ at $E$ ($E$ between $A$ and $B$) and let circumcircle of $ABD$ intersect $AC$ at $F$ ($F$ between $A$ and $C$). Let $O$ be the circumcenter of $AEF$. Prove that $OD$ bisects $\angle EDF$.

2017 CHMMC (Fall), 3

Tags:

Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes, (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight) a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?

2023 USAMTS Problems, 3

Tags: number theory

Lizzie and Alex are playing a game on the whiteboard. Initially, $n$ twos are written on the board. On a player’s turn they must either 1. change any single positive number to 0, or 2. subtract one from any positive number of positive numbers on the board. The game ends once all numbers are 0, and the last player who made a move wins. If Lizzie always plays first, find all $n$ for which Lizzie has a winning strategy.

2015 Iran MO (3rd round), 6

Tags: algebra , inequalities

$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that: $\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$

1968 Dutch Mathematical Olympiad, 4

Tags: geometry , reflection , construction

Given is a triangle $ABC$. A line $\ell$ passes through reflection wrt $BC$ changes into the line $\ell'$, $\ell'$ changes into $\ell''$ through reflection wrt $AC$ and $\ell''$ through reflection wrt $AB$ changes into $\ell'''$. Construct the line $\ell$ given that $\ell'''$ coincides with $\ell$.

1996 All-Russian Olympiad Regional Round, 8.7

Tags: number theory , algebra

Dunno wrote several different natural numbers on the board and divided (in his head) the sum of these numbers by their product. After this, Dunno erased the smallest number and divided (again in his mind) the amount of the remaining numbers by their product. The second result was $3$ times greater than the first. What number did Dunno erase?

2012 Mathcenter Contest + Longlist, 2 sl11

Tags: perfect square , number theory

Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares? [i](PP-nine)[/i]

2008 Mongolia Team Selection Test, 1

Tags: counting , derangement , pigeonhole principle , combinatorics

How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?

2023-24 IOQM India, 20

Tags: combinatorics

For any finite non empty set $X$ of integers, let $\max (X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that $$ \begin{aligned} & \max (A) \times|B|=12 ; \text { and } \\ & |A| \times \max (B)=11 \end{aligned} $$ and $N$ can be written as $100 a+b$ where $a, b$ are positive integers less than 100 , find $a+b$.

2010 Contests, 2

Tags: polynomial , ceiling function , number theory , prime number , algebra

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2022 HMNT, 8

Tags: algebra

Alice thinks of four positive integers $a\leq b\leq c\leq d$ satisfying $\{ab+cd,ac+bd,ad+bc\}=\{40,70,100\}$. What are all the possible tuples $(a,b,c,d)$ that Alice could be thinking of?

2019 SAFEST Olympiad, 3

Tags: algebra , polynomial

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

2009 Dutch IMO TST, 3

Tags: inequalities , 3-variable inequality , three variable inequality

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

2015 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics

We have $n$ integers $a_1, a_2,. . . , a_n$, not necessarily distinct, with sum $2S.$ An integer $k$ is called [i]separator[/i] if $k$ of the numbers can be chosen with sum equal to $S.$ What is the maximum possible number of separators?

2024 Iran MO (3rd Round), 3

Tags: combinatorics , graph theory

$m,n$ are given integer numbers such that $m+n$ is an odd number. Edges of a complete bipartie graph $K_{m,n}$ are labeled by ${-1,1}$ such that the sum of all labels is $0$. Prove that there exists a spanning tree such that the sum of the labels of its edges is equal to $0$.

2012 ELMO Shortlist, 2

Tags: inequalities

Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that \[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\] [i]Owen Goff.[/i]

1991 AIME Problems, 9

Tags: trigonometry

Suppose that $\sec x+\tan x=\frac{22}7$ and that $\csc x+\cot x=\frac mn,$ where $\frac mn$ is in lowest terms. Find $m+n.$

1962 Putnam, A4

Tags: inequalities , derivative

Assume that $|f(x)|\leq 1$ and $|f''(x)|\leq 1$ for all $x$ on an interval of length at least $2.$ Show that $|f'(x)|\leq 2 $ on the interval.

1998 Gauss, 5

Tags: gauss

If a machine produces $150$ items in one minute, how many would it produce in $10$ seconds? $\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 30$

2006 China Girls Math Olympiad, 3

Tags: number theory

Show that for any $i=1,2,3$, there exist infinity many positive integer $n$, such that among $n$, $n+2$ and $n+28$, there are exactly $i$ terms that can be expressed as the sum of the cubes of three positive integers.

2018 Saint Petersburg Mathematical Olympiad, 7

Tags: geometry

Points $A,B$ lies on the circle $S$. Tangent lines to $S$ at $A$ and $B$ intersects at $C$. $M$ -midpoint of $AB$. Circle $S_1$ goes through $M,C$ and intersects $AB$ at $D$ and $S$ at $K$ and $L$. Prove, that tangent lines to $S$ at $K$ and $L$ intersects at point on the segment $CD$.

2023 AMC 8, 10

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Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left? $\textbf{(A)}\ \frac{1}{12} \qquad \textbf{(B)}\ \frac{1}{6} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{5}{12}$

2021 Purple Comet Problems, 5

Tags: algebra

Ted is five times as old as Rosie was when Ted was Rosie's age. When Rosie reaches Ted's current age, the sum of their ages will be $72$. Find Ted's current age.