Olympiad problems

2024 AMC 12/AHSME, 13

Tags: system of equations

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$ What is the minimum possible value of $h+k$? $ \textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad $

1999 Austrian-Polish Competition, 6

Tags: algebra , system of equations

Solve in the nonnegative real numbers the system of equations $$\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\,\, for \,\,\,\, n = 1,2,..., 1999 \\\ x_0 = x_{1999} \end{cases}$$

2024 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , system of equations

Let $x$ and $y$ be positive real numbers satisfying the following system of equations: \[ \begin{cases} \sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\ \sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2 \end{cases} \] Find the maximum value of $x + y$.

1995 IMO Shortlist, 2

Tags: algebra , system of equations , equation

Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that \[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]

1995 Swedish Mathematical Competition, 3

Tags: algebra , system of equations , system

Let $a,b,x,y$ be positive numbers with $a+b+x+y < 2$. Given that $$\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases} $$ show that $a = x$ and $b = y$

1956 Czech and Slovak Olympiad III A, 1

Tags: trigonometric equation , system of equations , cosine , sine , algebra

Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that \begin{align*} \frac{\cos x}{\cos y}&=2\cos^2 y, \\ \frac{\sin x}{\sin y}&=2\sin^2 y. \end{align*}

1997 German National Olympiad, 4

Tags: system of equations , algebra

Find all real solutions $(x,y,z)$ of the system of equations $$\begin{cases} x^3 = 2y-1 \\y^3 = 2z-1\\ z^3 = 2x-1\end{cases}$$

2011 Romanian Masters In Mathematics, 2

Tags: algebra , polynomial , modular arithmetic , function , number theory , greatest common divisor , system of equations

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

2005 Estonia National Olympiad, 1

Tags: trigonometry , system of equations , algebra

Real numbers $x$ and $y$ satisfy the system of equalities $$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$ Prove that $\cos 2x = \cos 2y$.

1994 Tuymaada Olympiad, 7

Tags: diophantine equation , algebra , system of equations , relatively prime

Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$

2015 Abels Math Contest (Norwegian MO) Final, 1a

Tags: system of equations , algebra

Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\ y^2 + 4z^2 = 4xy \\ z^2 + 4x^2 = 4yz \end{cases}$

2009 Denmark MO - Mohr Contest, 2

Tags: algebra , system of equations , system

Solve the system of equations $$\begin{cases} \dfrac{1}{x+y}+ x = 3 \\ \\ \dfrac{x}{x+y}=2 \end{cases}$$

2013 Hanoi Open Mathematics Competitions, 14

Tags: algebra , system of equations

Solve the system of equations $\begin{cases} x^3+y = x^2+1\\ 2y^3+z=2y^2+1 \\ 3z^3+x=3z^2+1 \end{cases}$

III Soros Olympiad 1996 - 97 (Russia), 10.4

Tags: number theory , system of equations

Find natural $a, b, c, d$ that satisfy the system $$\begin{cases} ab+cd=34 \\ ac-bd=19 \end{cases}$$

2007 Federal Competition For Advanced Students, Part 2, 2

Tags: system of equations , algebra

Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds: $ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\ x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\ x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\ x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\ x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\ x_6x_1(1\minus{}x_2)\equal{}x_3x_4$

2025 JBMO TST - Turkey, 3

Tags: algebra , system of equations

Find all positive real solutions $(a, b, c)$ to the following system: $$ \begin{aligned} a^2 + \frac{b}{a} &= 8, \\ ab + c^2 &= 18, \\ 3a + b + c &= 9\sqrt{3}. \end{aligned} $$

2016 CMIMC, 3

Tags: algebra , system of equations , yuh.

Suppose $x$ and $y$ are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then $x^2+y^2$ can be written in the form $\sqrt[m]{n}$, where $m$ and $n$ are positive integers and $m$ is as small as possible. Find $m+n$.

1974 IMO, 1

Tags: combinatorics , game , algebra , system of equations

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

2003 AMC 12-AHSME, 24

Tags: ceiling function , analytic geometry , graphing lines , slope , function , algebra , system of equations

Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations \[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| \]has exactly one solution. What is the minimum value of $ c$? $ \textbf{(A)}\ 668 \qquad \textbf{(B)}\ 669 \qquad \textbf{(C)}\ 1002 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 2004$

2006 Czech-Polish-Slovak Match, 2

Tags: invariant , modular arithmetic , arithmetic sequence , algebra , system of equations , combinatorics

There are $n$ children around a round table. Erika is the oldest among them and she has $n$ candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which $n \ge 3$ is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?

1992 All Soviet Union Mathematical Olympiad, 564

Tags: system of equations , algebra

Find all real $x, y$ such that $$\begin{cases}(1 + x)(1 + x^2)(1 + x^4) = 1+ y^7 \\ (1 + y)(1 + y^2)(1 + y^4) = 1+ x^7 \end{cases}$$

1994 China Team Selection Test, 2

Tags: function , ratio , arithmetic sequence , algebra , system of equations , combinatorics

An $n$ by $n$ grid, where every square contains a number, is called an $n$-code if the numbers in every row and column form an arithmetic progression. If it is sufficient to know the numbers in certain squares of an $n$-code to obtain the numbers in the entire grid, call these squares a key. [b]a.) [/b]Find the smallest $s \in \mathbb{N}$ such that any $s$ squares in an $n-$code $(n \geq 4)$ form a key. [b]b.)[/b] Find the smallest $t \in \mathbb{N}$ such that any $t$ squares along the diagonals of an $n$-code $(n \geq 4)$ form a key.

2005 Putnam, B5

Tags: algebra , polynomial , calculus , derivative , system of equations

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that (a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically) and that (b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$ Show that $P=0$ identically.

2017 CMIMC Algebra, 7

Tags: algebra , system of equations

Let $a$, $b$, and $c$ be complex numbers satisfying the system of equations \begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*} Find $abc$.

2024 AMC 10, 23

Tags: system of equations

Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$? $ \textbf{(A) }212 \qquad \textbf{(B) }247 \qquad \textbf{(C) }258 \qquad \textbf{(D) }276 \qquad \textbf{(E) }284 \qquad $