Olympiad problems

2010 China Northern MO, 1

It is known that the sequence $\{a_n\}$ satisfies $a_1=2$, $a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}$, $(n \ge 2)$, find the general term of $a_n$.

2020 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequality , 3-variable inequality

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

PEN Q Problems, 7

Tags: algebra , polynomial , calculus , integration , number theory , relatively prime , rational root theorem

Let $f(x)=x^{n}+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

2011 Serbia JBMO TST, 1

Tags: combinatorics , algebra , number theory

A $tetromino$ is a figure made up of four unit squares connected by common edges. [List=i] [*] If we do not distinguish between the possible rotations of a tetromino within its plane, prove that there are seven distinct tetrominos. [*]Prove or disprove the statement: It is possible to pack all seven distinct tetrominos into $4\times 7$ rectangle without overlapping. [/list]

LMT Team Rounds 2010-20, B1

Tags: algebra

Four $L$s are equivalent to three $M$s. Nine $M$s are equivalent to fourteen $T$ s. Seven $T$ s are equivalent to two $W$ s. If Kevin has thirty-six $L$s, how many $W$ s would that be equivalent to?

2006 Kyiv Mathematical Festival, 3

Tags: symmetry , algebra

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Find all positive integers $a, b, c$ such that $3abc+11(a+b+c)=6(ab+bc+ac)+18.$

2017 Spain Mathematical Olympiad, 5

Tags: algebra , inequalities , maximum

Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of $$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$

2023 China Girls Math Olympiad, 6

Tags: algebra

Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of $$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$

2008 JBMO Shortlist, 4

Tags: algebra , system of equations

Find all triples $(x,y,z)$ of real numbers that satisfy the system $\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$

1996 All-Russian Olympiad Regional Round, 8.1

Tags: algebra

Ice cream costs $2000$ rubles. Petya has $$400^5 - 399^2\cdot (400^3 + 2\cdot 400^2 + 3\cdot 400 + 4)$$ rubles. Does Petya have enough money for ice cream?

2006 IberoAmerican Olympiad For University Students, 2

Tags: function , trigonometry , ratio , arithmetic sequence , algebra

Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation \[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\] has at least one real root.

2025 Kyiv City MO Round 1, Problem 5

Tags: inequalities , algebra

Real numbers $ a, b, c $ satisfy the following conditions: \[ 1000 < |a| < 2000, \quad 1000 < |b| < 2000, \quad 1000 < |c| < 2000, \] and \[ \frac{ab^2}{a+b} + \frac{bc^2}{b+c} + \frac{ca^2}{c+a} = 0. \] What are the possible values of the expression \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{a}? \] [i]Proposed by Vadym Solomka[/i]

1998 Vietnam Team Selection Test, 1

Tags: algebra , polynomial , calculus , integration , rational root theorem , number theory

Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

2021-IMOC, A2

Tags: algebra

For any positive integers $n$, find all $n$-tuples of complex numbers $(a_1,..., a_n)$ satisfying $$(x+a_1)(x+a_2)\cdots (x+a_n)=x^n+\binom{n}{1}a_1 x^{n-1}+\binom{n}{2}a_2^2 x^{n-2}+\cdots +\binom{n}{n-1} a_{n-1}^{n-1}+\binom{n}{n}a_n^n.$$ Proposed by USJL.

2009 Putnam, B4

Tags: algebra , polynomial , vector , calculus , integration , complex analysis

Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

MathLinks Contest 6th, 6.1

Tags: algebra , inequalities

Let $p > 1$ and let $a, b, c, d$ be positive numbers such that $$(a + b + c + d) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)= 16p^2.$$ Find all values of the ratio $ R =\frac{\max \{a, b, c, d\}}{\min \{a, b, c, d\}}$ (depending on the parameter $p$)

1977 Germany Team Selection Test, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

2015 Iran Team Selection Test, 1

Tags: polynomial , algebra

Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $$P(x)^3+Q(x)^3=x^{12}+1.$$

2019 Miklós Schweitzer, 7

Tags: algebra , complex analysis

Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.

2015 MMATHS, 3

Tags: number theory , algebra

Is there a number $s$ in the set $\{\pi,2\pi,3\pi,...,\} $ such that the first three digits after the decimal point of $s$ are $.001$? Fully justify your answer.

2022 LMT Spring, 4

Tags: algebra

Kevin runs uphill at a speed that is $4$ meters per second slower than his speed when he runs downhill. Kevin takes a total of $80$ seconds to run up and down a hill on one path. Given that the path is $300$ meters long (he travels $600$ meters total), find how long Kevin takes to run up the hill in seconds.

2004 Harvard-MIT Mathematics Tournament, 7

Tags: algebra , geometry

Farmer John is grazing his cows at the origin. There is a river that runs east to west $50$ feet north of the origin. The barn is $100$ feet to the south and $80$ feet to the east of the origin. Farmer John leads his cows to the river to take a swim, then the cows leave the river from the same place they entered and Farmer John leads them to the barn. He does this using the shortest path possible, and the total distance he travels is $d$ feet. Find the value of $d$.

2000 Switzerland Team Selection Test, 8

Tags: algebra , sum

Let $f(x) = \frac{4^x}{4^x+2}$ for $x > 0$. Evaluate $\sum_{k=1}^{1920}f\left(\frac{k}{1921}\right)$

2025 Romania National Olympiad, 3

Tags: floor function , algebra , bounding

Let $n \geq 2$ be a positive integer. Consider the following equation: \[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\] a) For $n=2$, solve the given equation in $\mathbb{R}$. b) Prove that, for any $n \geq 2$, the equation has at most $2$ real solutions.

2020 Korea Junior Math Olympiad, 5

Tags: algebra , inequalities

Let $a, b, c, d, e$ be real numbers satisfying the following conditions. \[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.