Olympiad problems

2015 BMT Spring, 4

Tags: sequence , algebra

Let $\{a_n\}$ be a sequence of real numbers with $a_1=-1$, $a_2=2$ and for all $n\ge3$, $$a_{n+1}-a_n-a_{n+2}=0.$$ Find $a_1+a_2+a_3+\ldots+a_{2015}$.

2024 Brazil National Olympiad, 5

Tags: algebra , quadratic , board , sequence , sum and product

Esmeralda chooses two distinct positive integers $a$ and $b$, with $b > a$, and writes the equation \[ x^2 - ax + b = 0 \] on the board. If the equation has distinct positive integer roots $c$ and $d$, with $d > c$, she writes the equation \[ x^2 - cx + d = 0 \] on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops. a) Show that Esmeralda can choose $a$ and $b$ such that she will write exactly 2024 equations on the board. b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?

2000 Croatia National Olympiad, Problem 1

Tags: algebra

Find all positive integer solutions $x,y,z$ such that $1/x +2/y - 3/z=1$

2015 Dutch IMO TST, 2

Tags: algebra , polynomial

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

2010 Contests, 1

Tags: function , vieta , algebra , functional equation

Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.

2023 BMT, 1

Tags: algebra

Lakshay chooses two numbers, $m$ and $n$, and draws two lines, $y = mx + 3$ and $y = nx + 23$. Given that the two lines intersect at $(20, 23)$, compute $m + n$.

2017 India PRMO, 5

Tags: geometric sequence , geometric progression , algebra , exponential

Let $u, v,w$ be real numbers in geometric progression such that $u > v > w$. Suppose $u^{40} = v^n = w^{60}$. Find the value of $n$.

2018 Spain Mathematical Olympiad, 1

Tags: algebra , number theory

Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.

2003 China Western Mathematical Olympiad, 1

Tags: calculus , integration , induction , function , quadratic , algebra

The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.

2023 Chile Classification NMO Seniors, 4

Tags: sfft , algebra

When writing the product of two three-digit numbers, the multiplication sign was omitted, forming a six-digit number. It turns out that the six-digit number is equal to three times the product. Find the six-digit number.

2021 Peru Cono Sur TST., P7

Tags: algebra

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

2011 Laurențiu Duican, 3

Tags: cauchy schwarz , algebra , polynom

Let $ n\ge 2 $ be a perfect square and let be $ n $ natural numbers $ m_1,m_2,\ldots ,m_n. $ Prove that if the polynom $$ X^2-\left( 1+ m_1^2+m_2^2+\cdots +m_n^2 \right) X+m_1m_2+m_2m_3+\cdots +m_{n-1}m_n +m_nm_1\in \mathbb{N} [X] $$ is reducible, then its two roots are perfect squares.

2019 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

A polynomial $f(x)$ of degree $2000$ is given. It's known that $f(x^2-1)$ has exactly $3400$ real roots while $f(1-x^2)$ has exactly $2700$ real roots. Prove that there exist two real roots of $f(x)$ such that the difference between them is less that $0.002$. [i](А. Солынин)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: algebra , inequalities , positive real

Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$. Prove that there are positive numbers $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$

2010 Finnish National High School Mathematics Competition, 3

Tags: polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

2022 Argentina National Olympiad, 6

Tags: algebra , polynomial

For every positive integer $n$, we consider the polynomial of real coefficients, of $2n+1$ terms, $$P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$$ where all coefficients are real numbers satisfying $100 \le a_i \le 101$ for $0 \le i \le 2n$. Find the smallest possible value of $n$ such that the polynomial can have at least one real root.

2000 National Olympiad First Round, 20

Tags: algebra , polynomial

For every real $x$, the polynomial $p(x)$ whose roots are all real satisfies $p(x^2-1)=p(x)p(-x)$. What can the degree of $p(x)$ be at most? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{There is no upper bound for the degree of } p(x) \qquad\textbf{(E)}\ \text{None} $

2009 Stanford Mathematics Tournament, 10

Tags: algebra

Evaluate $\sum_{n=2009}^{\infty} \frac{ {n \choose 2009}}{2^n}$

2003 Poland - Second Round, 3

Tags: algebra , polynomial , polynomial equation

Let $W(x) = x^4 - 3x^3 + 5x^2 - 9x$ be a polynomial. Determine all pairs of different integers $a$, $b$ satisfying the equation $W(a) = W(b)$.

2018 District Olympiad, 2

Tags: algebra , inequalities

Let $a,b,c \in [1, \infty)$. Prove that: $$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$

the 11th XMO, 10

Tags: complex number , algebra

Given $t\in\mathbb C$. Complex numbers $x,y,z$ satisfy that $|x|=|y|=|z|=1$ and $\frac{t}{y}=\frac{1}{x}+\frac{1}{z}$. Calculate $$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.$$

2007 Switzerland - Final Round, 7

Tags: inequalities , algebra , radical

Let $a, b, c$ be nonnegative real numbers with arithmetic mean $m =\frac{a+b+c}{3}$ . Provethat $$\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.$$

2023 Girls in Mathematics Tournament, 2

Tags: algebra , number theory

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.

2000 Harvard-MIT Mathematics Tournament, 8

Tags: algebra

Bobo the clown was juggling his spherical cows again when he realized that when he drops a cow is related to how many cows he started off juggling. If he juggles $1$, he drops it after $64$ seconds. When juggling $2$, he drops one after $55$ seconds, and the other $55$ seconds later. In fact, he was able to create the following table: [img]https://cdn.artofproblemsolving.com/attachments/1/0/554a9bace83b4b3595c6012dfdb42409465829.png[/img] He can only juggle up to $22$ cows. To juggle the cows the longest, what number of cows should he start off juggling? How long (in minutes) can he juggle for?

2011 Saudi Arabia BMO TST, 2

Tags: algebra , polynomial

Let $n$ be a positive integer. Prove that all roots of the equation $$x(x + 2) (x + 4 )... (x + 2n) + (x +1) (x + 3 )... (x + 2n - 1) = 0$$ are real and irrational.