Olympiad problems

2006 Brazil National Olympiad, 3

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that \[f(xf(y)+f(x)) = 2f(x)+xy\] for every reals $x,y$.

2022 MIG, 1

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In a certain store, all pencils cost the same amount of money. If three pencils can be bought for six dollars, what is the price of two pencils? $\textbf{(A) }\$ 3\qquad\textbf{(B) }\$ 3.5\qquad\textbf{(C) }\$ 4\qquad\textbf{(D) }\$4.5\qquad\textbf{(E) }\$ 5$

2015 Tuymaada Olympiad, 6

Tags: number theory

Is there sequence $(a_n)$ of natural numbers, such that differences $\{a_{n+1}-a_n\}$ take every natural value and only one time and differences $\{a_{n+2}-a_n\}$ take every natural value greater $2015$ and only one time ? [i]A. Golovanov[/i]

2017 Putnam, B2

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Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[N=a+(a+1)+(a+2)+\cdots+(a+k-1)\] for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?

2015 USAJMO, 4

Tags: function , arithmetic sequence

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)

2015 Putnam, A4

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For each real number $x,$ let \[f(x)=\sum_{n\in S_x}\frac1{2^n}\] where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x)\ge L$ for all $x\in [0,1)$? (As usual, $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z.$

2002 Moldova National Olympiad, 1

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Before going to vacation, each of the $ 7$ pupils decided to send to each of the $ 3$ classmates one postcard. Is it possible that each student receives postcards only from the classmates he has sent postcards?

2013 USAMTS Problems, 2

Tags: induction

Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.

2020 Polish Junior MO First Round, 6.

Tags: number theory , digit

Let $a$, $b$ $c$ be the natural numbers, such that every digit occurs exactly the same number of times in each of the numbers $a$, $b$, $c$. Is it possible that $a + b + c = 10^{1001}$? Justify your answer.

2003 Bulgaria National Olympiad, 3

Tags: induction , algebra

Given the sequence $\{y_n\}_{n=1}^{\infty}$ defined by $y_1=y_2=1$ and \[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\] find all integers $k$ such that every term of the sequence is a perfect square.

2019 Bulgaria National Olympiad, 1

Tags: inequality , inequalities

Let $f(x)=x^2+bx+1,$ where $b$ is a real number. Find the number of integer solutions to the inequality $f(f(x)+x)<0.$

1959 Kurschak Competition, 2

Tags: geometry , angle

The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

2013 USA Team Selection Test, 2

Tags: geometry , circumcircle , similar triangles , perpendicular bisector

Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.

2023 China Western Mathematical Olympiad, 7

Tags: algebra , inequality , number theory

For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

2020 Malaysia IMONST 1, 11

Tags: modulo , number theory

If we divide $2020$ by a prime $p$, the remainder is $6$. Determine the largest possible value of $p$.

2002 AMC 12/AHSME, 10

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How many different integers can be expressed as the sum of three distinct members of the set $ \{1, 4, 7, 10, 13, 16, 19\}$? $ \textbf{(A)}\ 13 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$

2009 Argentina National Olympiad, 2

Tags: number theory , divisor

A positive integer $n$ is [i]acceptable [/i] if the sum of the squares of its proper divisors is equal to $2n+4$ (a divisor of $n$ is [i]proper [/i] if it is different from $1$ and of $n$ ). Find all acceptable numbers less than $10000$,

2010 Polish MO Finals, 2

Tags: modular arithmetic , quadratic , trigonometry , number theory

Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.

2019 BMT Spring, Tie 3

Tags: combinatorics

Ankit, Bill, Charlie, Druv, and Ed are playing a game in which they go around shouting numbers in that order. Ankit starts by shouting the number $1$. Bill adds a number that is a factor of the number of letters in his name to Ankit’s number and shouts the result. Charlie does the same with Bill’s number, and so on (once Ed shouts a number, Ankit does the same procedure to Ed’s number, and the game goes on). What is the sum of all possible numbers that can be the $23$rd shout?

1987 AMC 12/AHSME, 26

Tags: probability

The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}.$ Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$? $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{2}{5} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{3}{4} $

2020 Jozsef Wildt International Math Competition, W20

Tags: asymptotics , real analysis , sequence

Let $p\in(0,1)$ and $a>0$ be real numbers. Determine the asymptotic behavior of the sequence $\{a_n\}_{n=1}^\infty$ defined recursively by $$a_1=a,a_{n+1}=\frac{a_n}{1+a_n^p},n\in\mathbb N$$ [i]Proposed by Arkady Alt[/i]

JOM 2015 Shortlist, C1

Tags: combinatorics

Baron and Peter are playing a game. They are given a simple finite graph $G$ with $n\ge 3$ vertex and $k$ edges that connects the vertices. First Peter labels two vertices A and B, and places a counter at A. Baron starts first. A move for Baron is move the counter along an edge. Peter's move is to remove an edge from the graph. Baron wins if he reaches $B$, otherwise Peter wins. Given the value of $n$, what is the largest $k$ so that Peter can always win?

2019 Auckland Mathematical Olympiad, 4

Tags: number theory

Find the smallest positive integer that cannot be expressed in the form $\frac{2^a - 2^b}{2^c - 2^d}$, where $a$, $ b$, $c$, $d$ are non-negative integers.

2015 Saudi Arabia JBMO TST, 2

Tags: inequalities , algebra

Let $a,b,c$ be positive real numbers. Prove that $$\frac{a}{\sqrt{(2a+b)(2a+c)}} +\frac{b}{\sqrt{(2b+c)(2b+a)}} +\frac{c}{\sqrt{(2c+a)(2c+b)}} \le 1 $$

Bangladesh Mathematical Olympiad 2020 Final, #9

Tags: number theory , number

You have 2020 piles of coins in front Of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. Guess a positive integer[b] k[/b], in which piles contain at least[b] k [/b]coins, take away exact[b] k[/b] coins from these piles. Find the [b]minimum number of turns[/b] you need to take way all of these coins?