Olympiad problems

2004 Romania National Olympiad, 1

Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$. [i]Cristinel Mortici[/i]

2016 Romanian Master of Mathematics Shortlist, A1

Tags: functional equation , algebra , functional equation in n

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

2012 Baltic Way, 4

Tags: quadratic , quadratic formula , algebra

Prove that for infinitely many pairs $(a,b)$ of integers the equation \[x^{2012} = ax + b\] has among its solutions two distinct real numbers whose product is 1.

1984 Balkan MO, 1

Tags: calculus , function , inequalities , n-variable inequality , algebra

Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$. Prove that: \[\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}\]

2011 India IMO Training Camp, 2

Tags: calculus , integration , algebra

Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.

1961 All-Soviet Union Olympiad, 1

Tags: sequence , algebra

Prove that for any three infinite sequences of natural numbers $(a_n)_{n\ge 1}$, $(b_n)_{n\ge 1}$, $(c_n)_{n\ge 1}$, there exist numbers $p$ and $q$ such that $a_p\ge a_q$, $b_p\ge b_q$ and $c_p\ge c_q$.

2001 Korea Junior Math Olympiad, 7

Tags: algebra , combinatorics , set

Finite set $\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}$ is given. If $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$, show that $$\sum_{i=1}^n |a_i-b_i|=n^2$$

2023 Baltic Way, 2

Tags: algebra

Let $a_1, a_2, \ldots, a_{2023}$ be positive reals such that $\sum_{i=1}^{2023}a_i^i=2023$. Show that $$\sum_{i=1}^{2023}a_i^{2024-i}>1+\frac{1}{2023}.$$

2025 Ukraine National Mathematical Olympiad, 11.5

Tags: polynomial , algebra

Initially, two constant polynomials are written on the board: $0$ and $1$. At each step, it is allowed to add $1$ to one of the polynomials and to multiply another one by the polynomial $45x + 2025$. Can the polynomials become equal at some point? [i]Proposed by Oleksii Masalitin[/i]

MMPC Part II 1958 - 95, 1958

Tags: number theory , geometry , 3d geometry , combinatorics , algebra

[b]p1.[/b] Show that $9x + 5y$ is a multiple of$ 17$ whenever $2x + 3y$ is a multiple of $17$. [b]p2.[/b] Express the five distinct fifth roots of $1$ in terms of radicals. [b]p3.[/b] Prove that the three perpendiculars dropped to the three sides of an equilateral triangle from any point inside the triangle have a constant sum. [b]p4.[/b] Find the volume of a sphere which circumscribes a regular tetrahedron of edge $a$. [b]p5.[/b] For any integer $n$ greater than $1$, show that $n^2-2n + 1$ is a factor at $n^{n-1}-1$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2023 Junior Balkan Team Selection Tests - Moldova, 12

Tags: algebra , inequality

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3. $ Prove that $$\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.$$

2014 European Mathematical Cup, 4

Tags: algebra , function

Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that for all $x,y\in{{\mathbb{R}}}$ holds $f(x^2)+f(2y^2)=(f(x+y)+f(y))(f(x-y)+f(y))$ [i]Proposed by Matija Bucić[/i]

2019 Mexico National Olympiad, 5

Tags: number theory , algebra

Let $a > b$ be relatively prime positive integers. A grashopper stands at point $0$ in a number line. Each minute, the grashopper jumps according to the following rules: [list] [*] If the current minute is a multiple of $a$ and not a multiple of $b$, it jumps $a$ units forward. [*] If the current minute is a multiple of $b$ and not a multiple of $a$, it jumps $b$ units backward. [*] If the current minute is both a multiple of $b$ and a multiple of $a$, it jumps $a - b$ units forward. [*] If the current minute is neither a multiple of $a$ nor a multiple of $b$, it doesn't move. [/list] Find all positions on the number line that the grasshopper will eventually reach.

2008 All-Russian Olympiad, 1

Tags: algebra , polynomial

Numbers $ a,b,c$ are such that the equation $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots.Prove that if $ \minus{} 2\leq a \plus{} b \plus{} c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$

2003 Polish MO Finals, 3

Tags: algebra , polynomial

Find all polynomials $W$ with integer coefficients satisfying the following condition: For every natural number $n, 2^n - 1$ is divisible by $W(n).$

2010 Canada National Olympiad, 5

Tags: polynomial , algebra

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.

2014 AIME Problems, 14

Tags: algebra , symmetry , polynomial , quadratic , sfft , linear algebra , contrived

Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.

2018 CMIMC Algebra, 4

Tags: algebra

2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero. At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their left and right. Suppose the new numbers on the ducklings sum to 1000. What is the maximum possible sum of the original numbers on all 2018 slips?

2022 Canadian Mathematical Olympiad Qualification, 4

Tags: algebra , polynomial , integer polynomial

For a non-negative integer $n$, call a one-variable polynomial $F$ with integer coefficients $n$-[i]good [/i] if: (a) $F(0) = 1$ (b) For every positive integer $c$, $F(c) > 0$, and (c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime. Show that there exist infinitely many non-constant polynomials that are not $n$-good for any $n$.

2018 Thailand TST, 2

Tags: algebra , function

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

2016 Hanoi Open Mathematics Competitions, 2

Tags: algebra , combinatorics

Given an array of numbers $A = (672, 673, 674, ..., 2016)$ on table. Three arbitrary numbers $a,b,c \in A$ are step by step replaced by number $\frac13 min(a,b,c)$. After $672$ times, on the table there is only one number $m$, such that (A): $0 < m < 1$ (B): $m = 1$ (C): $1 < m < 2$ (D): $m = 2$ (E): None of the above.

2022 CMWMC, R5

Tags: algebra , combinatorics , geometry , number theory

[u]Set 5[/u] [b]p13.[/b] An equiangular $12$-gon has side lengths that alternate between $2$ and $\sqrt3$. Find the area of the circumscribed circle of this $12$-gon. [b]p14.[/b] For positive integers $n$, let $\sigma(n)$ denote the number of positive integer factors of $n$. Then $\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64$. Let $S$ be the set of factors $k$ of $17280$ such that $\sigma(k) = 32$. If $p$ is the product of the elements of $S$, find $\sigma(p)$. [b]p15.[/b] How many odd $3$-digit numbers have exactly four $1$’s in their binary (base $2$) representation? For example, $225_{10} = 11100001_2$ would be valid. PS. You should use hide for answers.

2005 Regional Competition For Advanced Students, 3

Tags: algebra , analytic geometry , function

For which values of $ k$ and $ d$ has the system $ x^3\plus{}y^3\equal{}2$ and $ y\equal{}kx\plus{}d$ no real solutions $ (x,y)$?

1985 Tournament Of Towns, (084) T5

Tags: algebra , combinatorics , number theory , sequence , sum of digits

Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ? (S. Fomin , Leningrad)

1990 Swedish Mathematical Competition, 6

Tags: algebra , number theory

Find all positive integers $m, n$ such that $\frac{117}{158} > \frac{m}{n} > \frac{97}{131}$ and $n \le 500$.