Olympiad problems

2024 Bangladesh Mathematical Olympiad, P6

Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.

2012 Romania National Olympiad, 4

Tags: superior algebra , algebra , polynomial

[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients. [/color]

2020 Turkey Team Selection Test, 8

Tags: inequalities , algebra

Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$

2019 Estonia Team Selection Test, 3

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

Kvant 2019, M2552

Tags: sequence , number theory , algebra

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

2011 Dutch BxMO TST, 3

Tags: algebra , equation , absolute value

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

2019 Latvia Baltic Way TST, 3

Tags: algebra

All integers are written on an axis in an increasing order. A grasshopper starts its journey at $x=0$. During each jump, the grasshopper can jump either to the right or the left, and additionally the length of its $n$-th jump is exactly $n^2$ units long. Prove that the grasshopper can reach any integer from its initial position.

2012 District Olympiad, 1

Tags: equation , fractional part , integer part , floor , algebra

Solve in $ \mathbb{R} $ the equation $ [x]^5+\{ x\}^5 =x^5, $ where $ [],\{\} $ are the integer part, respectively, the fractional part.

2015 Putnam, B4

Tags: algebra , summation , geometric series , rip kent

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express \[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\] as a rational number in lowest terms.

2007 Germany Team Selection Test, 1

Tags: floor function , recurrence relation , sequence , periodic sequence , algebra

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Russian TST 2017, P1

Tags: algebra , square root

Prove that $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_{119}}$ is an integer, where \[a_n=2-\frac{1}{n^2+\sqrt{n^4+1/4}}.\]

2007 Today's Calculation Of Integral, 182

Tags: algebra , domain , function , integration , geometry , inequalities , calculus

Find the area of the domain of the system of inequality \[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]

1985 USAMO, 2

Tags: quadratic , biquadratic , algebra

Determine each real root of \[x^4-(2\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\] correct to four decimal places.

2016 China Team Selection Test, 1

Tags: algebra , inequalities

Let $n$ be an integer greater than $1$, $\alpha$ is a real, $0<\alpha < 2$, $a_1,\ldots ,a_n,c_1,\ldots ,c_n$ are all positive numbers. For $y>0$, let $$f(y)=\left(\sum_{a_i\le y} c_ia_i^2\right)^{\frac{1}{2}}+\left(\sum_{a_i>y} c_ia_i^{\alpha} \right)^{\frac{1}{\alpha}}.$$ If positive number $x$ satisfies $x\ge f(y)$ (for some $y$), prove that $f(x)\le 8^{\frac{1}{\alpha}}\cdot x$.

2014 Brazil Team Selection Test, 4

Tags: functional equation , function , algebra

Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

2019 Puerto Rico Team Selection Test, 3

Tags: algebra , inequalities

Find the largest value that the expression can take $a^3b + b^3a$ where $a, b$ are non-negative real numbers, with $a + b = 3$.

2001 Moldova National Olympiad, Problem 1

Tags: algebra , inequalities

Find all real solutions of the equation $$x^2+y^2+z^2+t^2=xy+yz+zt+t-\frac25.$$

2018 Indonesia MO, 5

Tags: algebra

Find all triples of reals $(x,y,z)$ satisfying: $$\begin{cases} \frac{1}{3} \min \{x,y\} + \frac{2}{3} \max \{x,y\} = 2017 \\ \frac{1}{3} \min \{y,z\} + \frac{2}{3} \max \{y,z\} = 2018 \\ \frac{1}{3} \min \{z,x\} + \frac{2}{3} \max \{z,x\} = 2019 \\ \end{cases}$$

1983 Polish MO Finals, 5

Tags: inequalities , algebra , vector , combinatorial geometry

On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$. Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$.

1995 Abels Math Contest (Norwegian MO), 1b

Tags: algebra

Prove that if $(x+\sqrt{x^2 +1})(y+\sqrt{y^2 +1})= 1$ for real numbers $x,y$, then $x+y = 0$.

2006 AMC 12/AHSME, 18

Tags: algebra , domain , cyclic function , function

The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and \[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x. \]What is the largest set of real numbers that can be in the domain of $ f$? $ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\ \textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$

2023 Malaysia IMONST 2, 4

Tags: algebra

Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.

1998 Bosnia and Herzegovina Team Selection Test, 2

Tags: algebra , inequality , inequalities

For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$

2002 Singapore MO Open, 2

Tags: algebra , sum , inequalities

Let $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ be real numbers between $1001$ and $2002$ inclusive. Suppose $ \sum_{i=1}^n a_i^2= \sum_{i=1}^n b_i^2$. Prove that $$\sum_{i=1}^n\frac{a_i^3}{b_i} \le \frac{17}{10} \sum_{i=1}^n a_i^2$$ Determine when equality holds.

2017 239 Open Mathematical Olympiad, 4

Tags: polynomial , algebra

A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)