Olympiad problems

1995 Baltic Way, 1

Find all triples $(x,y,z)$ of positive integers satisfying the system of equations \[\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}\]

2020 Moldova Team Selection Test, 5

Tags: system of equations , parameterization , trigonometry , algebra

Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$

2003 Switzerland Team Selection Test, 1

Tags: parameter , system of equations , algebra

Real numbers $x,y,a$ satisfy the equations $$x+y = x^3 +y^3 = x^5 +y^5 = a$$ Find all possible values of $a$.

2012 India Regional Mathematical Olympiad, 6

Tags: algebra , solve system , system of equations

Solve the system of equations for positive real numbers: $$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$

2016 Regional Olympiad of Mexico West, 5

Tags: algebra , system of equations

Determine all real solutions of the following system of equations: $$x+y^2=y^3$$ $$y+x^2=x^3$$

1945 Moscow Mathematical Olympiad, 097

Tags: algebra , system of equations , parameter

The system $\begin{cases} x^2 - y^2 = 0 \\ (x - a)^2 + y^2 = 1 \end{cases}$ generally has four solutions. For which $a$ the number of solutions of the system is equal to three or two?

1979 IMO Longlists, 50

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.

2023 Kyiv City MO Round 1, Problem 2

Tags: system of equations , algebra

For any given real $a, b, c$ solve the following system of equations: $$\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right.$$ [i]Proposed by Oleksiy Masalitin, Bogdan Rublov[/i]

1996 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: system of equations , algebra

The real numbers $x, y, z$, distinct in pairs satisfy $$\begin{cases} x^2=2 + y \\ y^2=2 + z \\ z^2=2 + x.\end{cases}$$ Find the possible values of $x^2 + y^2 + z^2$.

2014 Singapore Senior Math Olympiad, 2

Tags: algebra , system of equations

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: \[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\]

2004 USAMTS Problems, 2

Tags: algebra , system of equations

Find positive integers $a$, $b$, and $c$ such that \[\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{219+\sqrt{10080}+\sqrt{12600}+\sqrt{35280}}.\] Prove that your solution is correct. (Warning: numerical approximations of the values do not constitute a proof.)

1986 IMO Longlists, 25

Tags: polynomial , system of equations , calculus , algebra

Let real numbers $x_1, x_2, \cdots , x_n$ satisfy $0 < x_1 < x_2 < \cdots< x_n < 1$ and set $x_0 = 0, x_{n+1} = 1$. Suppose that these numbers satisfy the following system of equations: \[\sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0 \quad \text{where } i = 1, 2, . . ., n.\] Prove that $x_{n+1-i} = 1- x_i$ for $i = 1, 2, . . . , n.$

2014 Singapore Junior Math Olympiad, 4

Tags: algebra , system of equations

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: $$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$$

2018 Iran MO (1st Round), 14

Tags: system of equations , algebra

For how many integers $k$ does the following system of equations has a solution other than $a=b=c=0$ in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b),\\ b^2+c^2 = ka(b+c),\\ c^2+a^2=kb(c+a).\end{cases}\end{align*}

1933 Eotvos Mathematical Competition, 1

Tags: algebra , system , system of equations

Let $a, b,c$ and $d$ be rea] numbers such that $a^2 + b^2 = c^2 + d^2 = 1$ and $ac + bd = 0$. Determine the value of $ab + cd$.

1987 Greece Junior Math Olympiad, 4

Tags: algebra , system of equations

If $$x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1 \ \ with \ \ x,y,z\in \mathbb{R},$$ prove that at least one of $x,y,z$ is equal to zero.

2017 NZMOC Camp Selection Problems, 8

Tags: algebra , system , system of equations

Find all possible real values for $a, b$ and $c$ such that (a) $a + b + c = 51$, (b) $abc = 4000$, (c) $0 < a \le 10$ and $c \ge 25$.

2024 Argentina National Math Olympiad Level 3, 1

Tags: algebra , system of equations

Find the real numbers $a$, $b$, $c$ and $d$ that satisfy the following equations: $$\left \{\begin{matrix} a\cdot b+c+d & = & 6, \\ b\cdot c+d+a & = & 2, \\ c\cdot d+a+b & = & 5, \\ d\cdot a+b+c & = & 3. \end{matrix}\right .$$

2000 District Olympiad (Hunedoara), 1

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

[b]a)[/b] Solve the system $$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$ [b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $

1993 Denmark MO - Mohr Contest, 3

Tags: algebra , system , system of equations

Determine all real solutions $x,y$ to the system of equations $$\begin{cases} x^2 + y^2 = 1 \\ x^6 + y^6 = \dfrac{7}{16} \end{cases}$$

2019 Latvia Baltic Way TST, 16

Tags: algebra , system of equations , factorial

Determine all tuples of positive integers $(x, y, z, t)$ such that: $$ xyz = t!$$ $$ (x+1)(y+1)(z+1) = (t+1)!$$ holds simultaneously.

1935 Moscow Mathematical Olympiad, 010

Tags: algebra , parameter , system of equations

Solve the system $\begin{cases} x^2 + y^2 - 2z^2 = 2a^2 \\ x + y + 2z = 4(a^2 + 1) \\ z^2 - xy = a^2 \end{cases}$

2008 Bulgarian Autumn Math Competition, Problem 9.1

Tags: system of equations , algebra

Solve the system $\begin{cases} x^2y^2+|xy|=\frac{4}{9}\\ xy+1=x+y^2\\ \end{cases}$

2013 Hanoi Open Mathematics Competitions, 13

Tags: algebra , system of equations

Solve the system of equations $\begin{cases} xy=1 \\ \frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}=1\end{cases}$

2002 VJIMC, Problem 1

Tags: algebra , complex number , system of equations

Find all complex solutions to the system \begin{align*} (a+ic)^3+(ia+b)^3+(-b+ic)^3&=-6,\\ (a+ic)^2+(ia+b)^2+(-b+ic)^2&=6,\\ (1+i)a+2ic&=0.\end{align*}