Olympiad problems

1985 Austrian-Polish Competition, 4

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

1999 AMC 12/AHSME, 17

Tags: system of equations , algebra , polynomial

Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$? $ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad \textbf{(B)}\ x \plus{} 80 \qquad \textbf{(C)}\ \minus{}x \plus{} 118 \qquad \textbf{(D)}\ x \plus{} 118 \qquad \textbf{(E)}\ 0$

1940 Moscow Mathematical Olympiad, 058

Tags: parameter , system of equations , algebra

Solve the system $\begin{cases} (x^3 + y^3)(x^2 + y^2) = 2b^5 \\ x + y = b \end{cases}$ in $C$

2009 District Olympiad, 1

Tags: system of equations , algebra

Find all non-negative real numbers $x, y, z$ satisfying $x^2y^2 + 1 = x^2 + xy$, $y^2z^2 + 1 = y^2 + yz$ and $z^2x^2 + 1 = z^2 + xz$.

1998 Estonia National Olympiad, 4

Tags: system of equations , algebra

For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$ Find the value of the expression $x^2 + y^2+ z^2$.

2003 Czech-Polish-Slovak Match, 1

Tags: system of equations , algebra

Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*} \max\{1, x_1\} &= x_2 \\ \max\{2, x_2\} &= 2x_3 \\ &\cdots \\ \max\{n, x_n\} &= nx_1. \\ \end{align*}

2019 Philippine MO, 3

Tags: least common multiple , number theory , system of equations

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$.

2019 Saudi Arabia Pre-TST + Training Tests, 3.2

Tags: system of equations , algebra

Find all triples of real numbers $(x, y,z)$ such that $$\begin{cases} x^4 + y^2 + 4 = 5yz \\ y^4 + z^2 + 4 = 5zx \\ z^4 + x^2 + 4 = 5xy\end{cases}$$

1980 All Soviet Union Mathematical Olympiad, 292

Tags: algebra , system of equations , trigonometry

Find real solutions of the system : $$\begin{cases} \sin x + 2 \sin (x+y+z) = 0 \\ \sin y + 3 \sin (x+y+z) = 0\\ \sin z + 4 \sin (x+y+z) = 0\end{cases}$$

1994 China Team Selection Test, 2

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

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.

2017 Irish Math Olympiad, 2

Tags: system , system of equations , algebra

Solve the equations : $$\begin{cases} a + b + c = 0 \\ a^2 + b^2 + c^2 = 1\\a^3 + b^3 +c^3 = 4abc \end{cases}$$ for $ a,b,$ and $c. $

2018 Latvia Baltic Way TST, P15

Tags: system of equations , number theory , algebra

Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations $$\begin{cases} x+y+z=n\\ xyz = 2t^3. \end{cases}$$

1917 Eotvos Mathematical Competition, 1

Tags: diophantine equation , system of equations , diophantine , number theory

If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

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 Belarus Team Selection Test, 4

Tags: system of equations , algebra

Given nonzero real numbers a,b,c with $a+b+c=a^2+b^2+c^2=a^3+b^3+c^3$. ($*$) a) Find $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)$ b) Do there exist pairwise different nonzero $a,b,c$ satisfying ($*$)? D. Bazylev

1992 IMO Longlists, 33

Tags: linear algebra , system of equations , algebra

Let $a, b, c$ be positive real numbers and $p, q, r$ complex numbers. Let $S$ be the set of all solutions $(x, y, z)$ in $\mathbb C$ of the system of simultaneous equations \[ax + by + cz = p,\]\[ax2 + by2 + cz2 = q,\]\[ax3 + bx3 + cx3 = r.\] Prove that $S$ has at most six elements.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.7

Tags: system of equations , trigonometry , algebra

Solve the system of equations $$\begin{cases} \sin^3 x+\sin^4 y=1 \\ \cos^4 x+\cos^5 y =1\end{cases}$$

1966 IMO Shortlist, 62

Tags: algebra , system of equations

Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

2015 NIMO Summer Contest, 15

Tags: funky , system of equations , algebra

Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]

2016 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: system of equations , algebra , number theory

Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.

2015 Caucasus Mathematical Olympiad, 3

Tags: system of equations , algebra

Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?

2004 IberoAmerican, 1

Tags: system of equations , modular arithmetic , number theory , algebra

Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a\plus{}b$ and $ 201a\plus{}b$ are both perfect squares.

2014 Online Math Open Problems, 12

Tags: algebra , prob , relatively prime , system of equations , number theory

Let $a$, $b$, $c$ be positive real numbers for which \[ \frac{5}{a} = b+c, \quad \frac{10}{b} = c+a, \quad \text{and} \quad \frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$. [i]Proposed by Evan Chen[/i]

1978 IMO Shortlist, 16

Tags: trigonometric substitution , trigonometry , system of equations , algebra

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

1999 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt , system of equations , algebra

Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?