Olympiad problems

1989 USAMO, 5

Let $u$ and $v$ be real numbers such that \[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \] Determine, with proof, which of the two numbers, $u$ or $v$, is larger.

2021 Peru MO (ONEM), 1

Tags: combinatorics , algebra

[b]a)[/b] Determine if it's possible write $6$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers. [b]b)[/b] Determine if it's possible write $8$ positive rational numbers, pairwise distinct, in a circle such that each one is equal to the product of your [b]neighbor[/b] numbers.

2007 Junior Tuymaada Olympiad, 2

Tags: algebra , trinomial , quadratic trinomial

Two quadratic trinomials $ f (x) $ and $ g (x) $ differ from each other only by a permutation of coefficients. Could it be that $ f (x) \geq g (x) $ for all real $ x $?

1996 Romania National Olympiad, 1

Tags: binomial coefficient , algebra , sum

For $n ,p \in N^*$ , $ 1 \le p \le n$, we define $$ R_n^p = \sum_{k=0}^p (p-k)^n(-1)^k C_{n+1}^k $$ Show that: $R_n^{n-p+1} =R_n^p$ .

2015 Princeton University Math Competition, A6/B7

Tags: algebra

We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by \[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]

2014 Contests, 4

Tags: algebra , geometry , perimeter , maximum

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?

2020 USA TSTST, 7

Tags: polynomial , algebra , complex number

Find all nonconstant polynomials $P(z)$ with complex coefficients for which all complex roots of the polynomials $P(z)$ and $P(z) - 1$ have absolute value 1. [i]Ankan Bhattacharya[/i]

2018 Iran Team Selection Test, 1

Tags: function , functional equation , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions: a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$ b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval. [i]Proposed by Navid Safaei[/i]

Bangladesh Mathematical Olympiad 2020 Final, #5

Tags: floor function , algebra

For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$.

1999 Bundeswettbewerb Mathematik, 1

Tags: number theory , combinatorics , algebra , pigeonhole principle

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts. (The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)

2000 Tournament Of Towns, 1

Tags: algebra , equation

Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$ (RM Kuznec)

2020 APMO, 5

Tags: number theory , algorithm , algebra

Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.

1966 IMO Longlists, 48

Tags: equation , root , parametric equation , number theory , algebra

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

2018 CMIMC Individual Finals, 3

Tags: algebra , polynomial

Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$.

2011 Switzerland - Final Round, 4

Tags: algebra , function

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true. [i](Swiss Mathematical Olympiad 2011, Final round, problem 4)[/i]

1986 India National Olympiad, 1

Tags: algebra

A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?

2019 LIMIT Category C, Problem 11

Tags: algebra

Let $$x=\frac1{1\cdot2}-\frac1{2\cdot3}+\frac1{3\cdot4}-\ldots$$Then $e^{x+1}$ is

1971 IMO Longlists, 31

Tags: polynomial , equation , algebra , root

Determine whether there exist distinct real numbers $a, b, c, t$ for which: [i](i)[/i] the equation $ax^2 + btx + c = 0$ has two distinct real roots $x_1, x_2,$ [i](ii)[/i] the equation $bx^2 + ctx + a = 0$ has two distinct real roots $x_2, x_3,$ [i](iii)[/i] the equation $cx^2 + atx + b = 0$ has two distinct real roots $x_3, x_1.$

2003 Estonia National Olympiad, 1

Tags: algebra

Juhan is touring in Europe. He stands on a highway and watches cars. There are three cars driving along the highway at constant speeds: an Opel and a Trabant in one direction and a Mercedes in the opposite direction. At the moment when the Trabant passes Juhan, the Opel and the Mercedes lie at equal distances from him in opposite directions. At the moment when the Mercedes passes Juhan, the Opel and the Trabant lie at equal distances from him in opposite directions. Prove that at the moment when the Opel passes Juhan, also the Mercedes and the Trabant lie at equal distances from him in opposite directions.

2020 Greece National Olympiad, 1

Tags: functional equation , functional , algebra , polynomial , polynomial equation

Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$

2022 Kosovo National Mathematical Olympiad, 4

Tags: inequalities , algebra

Let $a,b$ and $c$ be positive real numbers such that $a+b+c+3abc\geq (ab)^2+(bc)^2+(ca)^2+3$. Show that the following inequality hold, $$\frac{a^3+b^3+c^3}{3}\geq\frac{abc+2021}{2022}.$$

2003 Greece National Olympiad, 2

Tags: algebra

Find all real solutions of the system \[\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}\]

2008 Princeton University Math Competition, A7

Tags: algebra

Suppose $x^9 = 1$ but $x^3 \ne 1$. Find a polynomial of minimal degree equal to $\frac{1}{1+x}$ .

2013 HMNT, 1

Tags: algebra

Two cars are driving directly towards each other such that one is twice as fast as the other. The distance between their starting points is $4$ miles. When the two cars meet, how many miles is the faster car from its starting point?

2017 China Team Selection Test, 4

Tags: polynomial , algebra

Show that there exists a degree $58$ monic polynomial $$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$ such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.