Olympiad problems

2018 Mathematical Talent Reward Programme, SAQ: P 2

Tags: polynomial , algebra

$P(x)$ is polynomial with real coefficients such that $\forall n \in \mathbb{Z}, P(n) \in \mathbb{Z}$. Prove that every coefficients of $P(x)$ is rational numbers.

2014 Costa Rica - Final Round, 5

Tags: algebra , functional equation , functional , recurrence relation

Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$

2010 Germany Team Selection Test, 3

Tags: function , inequalities , algebra , functional inequality

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

1945 Moscow Mathematical Olympiad, 104

Tags: nested radical , radical , trigonometry , algebra

The numbers $a_1, a_2, ..., a_n$ are equal to $1$ or $-1$. Prove that $$2 \sin \left(a_1+\frac{a_1a_2}{2}+\frac{a_1a_2a_3}{4}+...+\frac{a_1a_2...a_n}{2^{n-1}}\right)\frac{\pi}{4}=a_1\sqrt{2+a_2\sqrt{2+a_3\sqrt{2+...+a_n\sqrt2}}}$$ In particular, for $a_1 = a_2 = ... = a_n = 1$ we have $$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$

2020 Thailand TST, 4

Tags: algebra , inequalities

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2022 Harvard-MIT Mathematics Tournament, 1

Tags: algebra

Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2-bx+c=0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.

2016 China Western Mathematical Olympiad, 6

Tags: inequalities , algebra

Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers ,$S_k= \sum\limits_{i=1}^{k}a_i $ $(1\le k\le n)$.Prove that$$\sum\limits_{i=1}^{n}\left(a_iS_i\sum\limits_{j=i}^{n}a^2_j\right)\le \sum\limits_{i=1}^{n}\left(a_iS_i\right)^2$$

2016 BMT Spring, 9

Tags: calculus , algebra

Suppose $p''(x) = 4x^2 + 4x + 2$ where $$p(x) = a_0 + a_1(x - 1) + a_2(x -2)^2 + a_3(x- 3)^4 + a_4(x-4)^4.$$ We have $p'(-3) = -24$ and $p(x)$ has the unique property that the sum of the third powers of the roots of $p(x)$ is equal to the sum of the fourth powers of the roots of $p(x)$ . Find $a_0$.

2024 Indonesia Regional, 1

Tags: inequality , inequalities , algebra

Given a real number $C\leqslant 2$. Prove that for every positive real number $x,y$ with $xy=1$, the following inequality holds: \[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\] [i]Proposed by Fajar Yuliawan, Indonesia[/i]

2025 Junior Macedonian Mathematical Olympiad, 4

Tags: inequalities , algebra , symmetric

Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality \[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\] When does the equality hold?

2022 Iran Team Selection Test, 12

Tags: algebra , interval , functional equation

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi

2024 District Olympiad, P1

Tags: algebra , function

Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that :$$(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.$$

2012 China Team Selection Test, 2

Tags: pigeonhole principle , floor function , ceiling function , function , algebra , difference of squares , inequalities

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

2016 Tournament Of Towns, 6

Tags: combinatorial game , algebra , polynomial , number theory , game theory , combinatorics

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

2007 Romania Team Selection Test, 4

Tags: algebra , polynomial , modular arithmetic , induction , number theory

i) Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime. ii) Find all polynomials $f(X) \in \mathbb{Z}[X]$, such that there exist $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, $| f(p) |$ is also prime. [i]Dan Schwarz[/i]

1990 Tournament Of Towns, (243) 1

Tags: algebra , sum

For every natural number $n$ prove that $$\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left( \frac{1}{n-1} + \frac12 \right)^2+ \left( \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)$$ (S. Manukian, Yerevan)

2021 Winter Stars of Mathematics, 1

Tags: algebra , inequality

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

2015 Romania National Olympiad, 2

Tags: function , algebra , polynomial , quadratic

A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $ Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $

1995 Tuymaada Olympiad, 7

Tags: functional equation , continuous function , algebra

Find a continuous function $f(x)$ satisfying the identity $f(x)-f(ax)=x^n-x^m$, where $n,m\in N , 0<a<1$

2023 Bangladesh Mathematical Olympiad, P10

Tags: algebra , polynomial , inequalities

Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$, where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$

2017 Harvard-MIT Mathematics Tournament, 10