Olympiad problems

PEN E Problems, 39

Let $c$ be a nonzero real number. Suppose that $g(x)=c_0x^r+c_1x^{r-1}+\cdots+c_{r-1}x+c_r$ is a polynomial with integer coefficients. Suppose that the roots of $g(x)$ are $b_1,\cdots,b_r$. Let $k$ be a given positive integer. Show that there is a prime $p$ such that $p>\max(k,|c|,|c_r|)$, and moreover if $t$ is a real number between $0$ and $1$, and $j$ is one of $1,\cdots,r$, then \[|(\text{ }c^r\text{ }b_j\text{}g(tb_j)\text{ })^pe^{(1-t)b}|<\dfrac{(p-1)!}{2r}.\] Furthermore, if \[f(x)=\dfrac{e^{rp-1}x^{p-1}(g(x))^p}{(p-1)!}\] then \[\left|\sum_{j=1}^r\int_0^1 e^{(1-t)b_j}f(tb_j)dt\right|\leq \dfrac{1}{2}.\]

2004 Croatia Team Selection Test, 1

Tags: quadratic , algebra , number theory

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

2008 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: inequalities , circumcircle , algebra , logarithm

If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]

2004 Baltic Way, 2

Tags: algebra , polynomial , inequalities , search

Let $ P(x)$ be a polynomial with a non-negative coefficients. Prove that if the inequality $ P\left(\frac {1}{x}\right)P(x)\geq 1$ holds for $ x \equal{} 1$, then this inequality holds for each positive $ x$.

2023 BMT, 7

Tags: algebra

Nikhil constructs a list of all polynomial pairs $(a(x), b(x))$ with real coefficients such that $a(x)$ has higher degree than $b(x)$ and $a(x)^2 + b(x)^2 = x^{10} + 1$. Danielle takes Nikhil’s list and adds all polynomial pairs that satisfy the same conditions but have complex coefficients. If Nikhil’s original list had $N$ pairs and Danielle added $D$ pairs, compute $D - N$.

2005 Thailand Mathematical Olympiad, 14

Tags: algebra , functional equation , functional

A function $f : Z \to Z$ is given so that $f(m + n) = f(m) + f(n) + 2mn - 2548$ for all positive integers $m, n$. Given that $f(2548) = -2548$, find the value of $f(2)$.

2003 Tuymaada Olympiad, 4

Tags: algebra , polynomial , calculus , integration , number theory

Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$ [i]Proposed by F. Petrov[/i] [hide="For those of you who liked this problem."] Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]

1984 IMO Longlists, 27

Tags: function , algebra

The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and \[f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)\] for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.

2009 Puerto Rico Team Selection Test, 5

Tags: algebra , number theory

The [i]weird [/i] mean of two numbers $ a$ and $ b$ is defined as $ \sqrt {\frac {2a^2 + 3b^2}{5}}$. $ 2009$ positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these $ 2009$ numbers must be equal.

2019 Hong Kong TST, 2

Tags: combinatorics , number theory , algebra , geometry , projective geometry

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

2014 Tajikistan Team Selection Test, 1

Tags: polynomial , algebra

Given the polynomial $p(x) = x^2 + x - 70$, do there exist integers $0<m<n$, so that $p(m)$ is divisible by $n$ and $p(m+1)$ is divisible by $n+1$? [i]Proposed by Nairy Sedrakyan[/i]

1978 Vietnam National Olympiad, 2

Tags: algebra , system of equations , parameterization

Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.

2013 China Northern MO, 7

Tags: algebra , sequence

Suppose that $\{a_n\}$ is a sequence such that $a_{n+1}=(1+\frac{k}{n})a_{n}+1$ with $a_{1}=1$.Find all positive integers $k$ such that any $a_n$ be integer.

2010 Contests, 1

Tags: pigeonhole principle , system of equations , algebra

Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations \[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]

2022-23 IOQM India, 4

Tags: algebra , diophantine

Starting with a positive integer $M$ written on the board , Alice plays the following game: in each move, if $x$ is the number on the board, she replaces it with $3x+2$.Similarly, starting with a positive integer $N$ written on the board, Bob plays the following game: in each move, if $x$ is the number on the board, he replaces it with $2x+27$.Given that Alice and Bob reach the same number after playing $4$ moves each, find the smallest value of $M+N$

1995 IMO Shortlist, 3

Tags: inequalities , n-variable inequality , algebra

Let $ n$ be an integer, $ n \geq 3.$ Let $ a_1, a_2, \ldots, a_n$ be real numbers such that $ 2 \leq a_i \leq 3$ for $ i \equal{} 1, 2, \ldots, n.$ If $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n,$ prove that \[ \frac{a^2_1 \plus{} a^2_2 \minus{} a^2_3}{a_1 \plus{} a_2 \minus{} a_3} \plus{} \frac{a^2_2 \plus{} a^2_3 \minus{} a^2_4}{a_2 \plus{} a_3 \minus{} a_4} \plus{} \ldots \plus{} \frac{a^2_n \plus{} a^2_1 \minus{} a^2_2}{a_n \plus{} a_1 \minus{} a_2} \leq 2s \minus{} 2n.\]

2005 Pan African, 2

Tags: algebra , polynomial

Let $S$ be a set of integers with the property that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. If $0$ and $1000$ are elements of $S$, prove that $-2$ is also an element of $S$.

2010 IMC, 5

Tags: function , algebra , polynomial , vector , analytic geometry , real analysis , number theory

Suppose that for a function $f: \mathbb{R}\to \mathbb{R}$ and real numbers $a<b$ one has $f(x)=0$ for all $x\in (a,b).$ Prove that $f(x)=0$ for all $x\in \mathbb{R}$ if \[\sum^{p-1}_{k=0}f\left(y+\frac{k}{p}\right)=0\] for every prime number $p$ and every real number $y.$

2024 Polish MO Finals, 5

Tags: inequalities , algebra

We are given an integer $n \ge 2024$ and a sequence $a_1,a_2,\dots,a_{n^2}$ of real numbers satisfying \[\vert a_k-a_{k-1}\vert \le \frac{1}{k} \quad \text{and} \quad \vert a_1+a_2+\dots+a_k\vert \le 1\] for $k=2,3,\dots,n^2$. Show that $\vert a_{n(n-1)}\vert \le \frac{2}{n}$. [i]Note: Proving $\vert a_{n(n-1)}\vert \le \frac{75}{n}$ will be rewarded partial points.[/i]

2025 Poland - Second Round, 6

Tags: inequalities , algebra

Let $1\le k\le n$. Suppose that the sequence $a_1, a_2, \ldots, a_n$ satisfies $0\le a_1 \le a_2 \le \ldots \le a_k$ and $0 \le a_n \le a_{n-1} \le \ldots \le a_k$. The sequence $b_1, b_2, \ldots, b_n$ is the nondecreasing permutation of $a_1, a_2, \ldots, a_n$. Prove that \[\sum_{i=1}^n \sum_{j=1}^n (j-i)^2a_ia_j \le \sum_{i=1}^n \sum_{j=1}^n (j-i)^2b_ib_j \]

1987 Greece Junior Math Olympiad, 3

Tags: algebra , polynomial

Find real $a,b$ such that polynomial $P(x)=x^{n+1}+ax+b$ to be divisible by $(x-1)^2$. Then find the quotient $P(x):(x-1)^2 , n\in \mathbb{N}^*$

2016 Israel Team Selection Test, 2

Tags: algebra , functional equation

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying (for all $x,y \in \mathbb{R}$): $f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)$.

2017 Princeton University Math Competition, B2

Tags: algebra , polynomial

Let $a_1(x), a_2(x)$, and $a_3(x)$ be three polynomials with integer coefficients such that every polynomial with integer coefficients can be written in the form $p_1(x)a_1(x) + p_2(x)a_2(x) + p_3(x)a_3(x)$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients. Show that every polynomial is of the form $p_1(x)a_1(x)^2 + p_2(x)a_2(x)^2 + p_3(x)a_3(x)^2$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients.

1988 Romania Team Selection Test, 13

Tags: quadratic , algebra

Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$, $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$. Prove that $(y,x)$ is a solution of the equation \[ |y^2 - axy - x^2 | = 1 \] if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$. [i]Serban Buzeteanu[/i]

2022 Kyiv City MO Round 1, Problem 2

Tags: algebra , inequalities

For any reals $x, y$, show the following inequality: $$\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20$$ [i](Proposed by Bogdan Rublov)[/i]