Olympiad problems

2004 Purple Comet Problems, 21

Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$.

PEN C Problems, 2

Tags: number theory , system of equations , gauss , quadratic residue , modular arithmetic , algebra

The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

2019 Purple Comet Problems, 18

Tags: algebra , system of equations

Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$.

1937 Moscow Mathematical Olympiad, 032

Tags: parameter , system of equations , algebra

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

2021 Baltic Way, 5

Tags: algebra , system of equations

Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$. Find all possible values of $x+y$.

1994 AIME Problems, 7

Tags: system of equations , geometry , quadratic , linear equation , analytic geometry , algebra

For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

2010 India National Olympiad, 3

Tags: algebra , system of equations , inequalities

Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations: \[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\] \[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]

2024 Polish MO Finals, 3

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

Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying \[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]

2007 Vietnam National Olympiad, 1

Tags: algebra , system of equations

Solve the system of equations: $\{\begin{array}{l}(1+\frac{12}{3x+y}).\sqrt{x}=2\\(1-\frac{12}{3x+y}).\sqrt{y}=6\end{array}$

1983 All Soviet Union Mathematical Olympiad, 352

Tags: system of equations , algebra

Find all the solutions of the system $$\begin{cases} y^2 = x^3 - 3x^2 + 2x \\ x^2 = y^3 - 3y^2 + 2y \end{cases}$$

2016 Vietnam National Olympiad, 1

Tags: algebra , system of equations

Solve the system of equations $\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1\quad\quad (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}$.

2000 Estonia National Olympiad, 3

Tags: algebra , system of equations

Prove that if the numbers $a, b, c, d$ satisfy the system of equations $$\begin{cases} a^2+b^2=2cd \\ b^2+c^2=2da \\ c^2+d^2=2ab \end{cases}$$ then $a=b=c=d$.

2024 AIME, 4

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

Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations: $$\log_2\left({x \over yz}\right) = {1 \over 2}$$ $$\log_2\left({y \over xz}\right) = {1 \over 3}$$ $$\log_2\left({z \over xy}\right) = {1 \over 4}$$ Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is ${m \over n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$

2009 Ukraine National Mathematical Olympiad, 1

Tags: algebra , system of equations

Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

1939 Moscow Mathematical Olympiad, 043

Tags: algebra , parameter , system of equations

Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\ x + y+ z = 2b \\ x^2 + y^2-z^2 = b^2 \end{cases}$ in $C$

1956 Czech and Slovak Olympiad III A, 1

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

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*}

2011 South africa National Olympiad, 2

Tags: algebra , system of equations

Suppose that $x$ and $y$ are real numbers that satisfy the system of equations $2^x-2^y=1$ $4^x-4^y=\frac{5}{3}$ Determine $x-y$

2013 Hanoi Open Mathematics Competitions, 14

Tags: system of equations , algebra

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

1979 IMO Shortlist, 18

Tags: number theory , additive number theory , additive combinatorics , system of equations

Let $m$ positive integers $a_1, \dots , a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \dots , b_n$ such that all sums of distinct $b_k$’s are distinct and all $a_i \ (i \leq m)$ occur among them.

2008 Regional Competition For Advanced Students, 2

Tags: algebra , system of equations

For a real number $ x$ is $ [x]$ the next smaller integer to $ x$, that is the integer $ g$ with $ g\leqq<g+1$, and $ \{x\}=x-[x]$ is the “decimal part” of $ x$. Determine all triples $ (a,b,c)$ of real numbers, which fulfil the following system of equations: \[ \{a\}+[b]+\{c\}=2,9\]\[ \{b\}+[c]+\{a\}=5,3\]\[\{c\}+[a]+\{b\}=4,0\]

2013 Hanoi Open Mathematics Competitions, 14

Tags: algebra , system of equations

Solve the system of equations $\begin{cases} x^3+\frac13 y=x^2+x -\frac43 \\ y^3+\frac14 z=y^2+y -\frac54 \\ z^3+\frac15 x=z^2+z -\frac65 \end{cases}$

2000 Romania National Olympiad, 2

Tags: algebra , system of equations

The negative real numbers $x, y, z, t$ satisfy simultaneously equalities, $$x + y + z = t, \,\,\,\,\frac{1}{x}+ \frac{1}{y}+\frac{1}{z}= \frac{1}{t}, \\,\,\,\, x^3 + y^3 + z^3 = 1000^3$$ Compute $x + y + z + t$.

1988 Federal Competition For Advanced Students, P2, 4

Tags: algebra , system of equations

Let $ a_{ij}$ be nonnegative integers such that $ a_{ij}\equal{}0$ if and only if $ i>j$ and that $ \displaystyle\sum_{j\equal{}1}^{1988}a_{ij}\equal{}1988$ holds for all $ i\equal{}1,...,1988$. Find all real solutions of the system of equations: $ \displaystyle\sum_{j\equal{}1}^{1988} (1\plus{}a_{ij})x_j\equal{}i\plus{}1, 1 \le i \le 1988$.

2018 Hanoi Open Mathematics Competitions, 8

Tags: system of equations , algebra

Let $a,b, c$ be real numbers with $a+b+c = 2018$. Suppose $x, y$, and $z$ are the distinct positive real numbers which are satisfied $a = x^2 - yz - 2018, b = y^2 - zx - 2018$ , and $c = z^2 - xy - 2018$. Compute the value of the following expression $P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$

2019 Czech and Slovak Olympiad III A, 1

Tags: algebra , system of equations

Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that \begin{align*} x^2-yz &= |y-z|+1, \\ y^2-zx &= |z-x|+1, \\ z^2-xy &= |x-y|+1. \end{align*}