Olympiad problems

2002 AIME Problems, 7

The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $ x, y,$ and $ r$ with $ |x| > |y|,$ \[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots \] What are the first three digits to the right of the decimal point in the decimal representation of $ \left(10^{2002} \plus{} 1\right)^{10/7}?$

2009 Romania Team Selection Test, 1

Tags: algebra , function , domain , geometric transformation , rotation , geometry

Given two (identical) polygonal domains in the Euclidean plane, it is not possible in general to superpose the two using only translations and rotations. Prove that this can however be achieved by splitting one of the domains into a finite number of polygonal subdomains which then fit together, via translations and rotations in the plane, to recover the other domain.

Russian TST 2020, P1

Tags: algebra , sequence

Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of \[ \left|1-\sum_{i \in X} a_{i}\right| \] is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that \[ \sum_{i \in X} b_{i}=1. \]

2014 Saint Petersburg Mathematical Olympiad, 3

Tags: number theory , combinatorics , algebra , inequalities

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$

2014 Contests, 1

Tags: inequalities , algebra

Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.

2008 Federal Competition For Advanced Students, P1, 2

Tags: algebra , system of equations

Given $a \in R^{+}$ and an integer $n > 4$ determine all n-tuples ($x_1, ...,x_n$) of positive real numbers that satisfy the following system of equations: $\begin {cases} x_1x_2(3a-2x_3) = a^3\\ x_2x_3(3a-2x_4) = a^3\\ ...\\ x_{n-2}x_{n-1}(3a-2x_n) = a^3\\ x_{n-1}x_n(3a-2x_1) = a^3 \\ x_nx_1(3a-2x_2) = a^3 \end {cases}$ .

2010 USAMO, 3

Tags: inequalities , ceiling function , algebra

The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.

2012 Kazakhstan National Olympiad, 1

Tags: function , algebra

Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.

1987 Swedish Mathematical Competition, 1

Tags: algebra , sum

Sixteen real numbers are arranged in a magic square of side $4$ so that the sum of numbers in each row, column or main diagonal equals $k$. Prove that the sum of the numbers in the four corners of the square is also $k$.

2024 AMC 12/AHSME, 15

Tags: algebra , those who know

The roots of $x^3 + 2x^2 - x + 3$ are $p, q,$ and $r.$ What is the value of \[(p^2 + 4)(q^2 + 4)(r^2 + 4)?\] $\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144$

2016 Hanoi Open Mathematics Competitions, 14

Tags: algebra , radical , natural

Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number.

2020 BMT Fall, 2

Tags: algebra

Let $m$ be the answer to this question. What is the value of $2m - 5$?

2002 District Olympiad, 1

Tags: algebra

Let $x, y, z$ be positive real numbers such that $xyz(x+y+z) = 1$. Show that the following equality holds: $$\sqrt{\left( x^2+\frac{1}{y^2}\right)\left( y^2+\frac{1}{z^2}\right)\left( z^2+\frac{1}{x^2}\right)}=(x+y)(y+z)(z+x)$$ Find some numbers $x ,y ,z$ which satisfy the given property.

2007 IMC, 5

Tags: function , modular arithmetic , algebra , polynomial , induction

Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.

2015 Caucasus Mathematical Olympiad, 2

Tags: algebra , inequalities , trinomial

The equation $(x+a) (x+b) = 9$ has a root $a+b$. Prove that $ab\le 1$.

2013 Bangladesh Mathematical Olympiad, 4

Tags: algebra

Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.

2009 Poland - Second Round, 3

Tags: vector , linear algebra , algebra

For every integer $n\ge 3$ find all sequences of real numbers $(x_1,x_2,\ldots ,x_n)$ such that $\sum_{i=1}^{n}x_i=n$ and $\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n$, where $x_0=x_n$ and $x_{n+1}=x_1$.

2022-23 IOQM India, 15

Tags: algebra , inequalities

Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression \\ $\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$\\. \\ If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$, find the value of $m+n$.

2005 AIME Problems, 8

Tags: vieta , algebra , polynomial , number theory , logarithm

The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

1997 Vietnam Team Selection Test, 2

Tags: algebra , logarithm

Find all pairs of positive real numbers $ (a, b)$ such that for every $ n \in\mathbb{N}^*$ and every real root $ x_n$ of the equation $ 4n^2x \equal{} \log_2(2n^2x \plus{} 1)$ we always have $ a^{x_n} \plus{} b^{x_n} \ge 2 \plus{} 3x_n$.

2005 Spain Mathematical Olympiad, 1

Tags: algebra , number theory

Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$ has an integer solution.

1966 All Russian Mathematical Olympiad, 077

Tags: algebra

Given the numbers $a_1, a_2, ... , a_n$ such that $$0\le a_1\le a_2\le 2a_1 , a_2\le a_3\le 2a_2 , ... , a_{n-1}\le a_n\le 2a_{n-1}$$ Prove that in the sum $s=\pm a1\pm a2\pm ...\pm a_n$ You can choose appropriate signs to make $0\le s\le a_1$.

2009 Princeton University Math Competition, 1

Tags: ratio , quadratic , greatest common divisor , algebra , quadratic formula , number theory , relatively prime

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.

1972 Yugoslav Team Selection Test, Problem 1

Tags: algebra , complex number

Given non-zero real numbers $u,v,w,x,y,z$, how many different possibilities are there for the signs of these numbers if $$(u+ix)(v+iy)(w+iz)=i?$$

2010 Kosovo National Mathematical Olympiad, 1

Tags: algebra

Solve the inequation $\sqrt {3-x}-\sqrt {x+1}>\frac {1}{2}$.