Olympiad problems

2016 Regional Olympiad of Mexico West, 5

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

1979 IMO Longlists, 44

Tags: diophantine equation , algebra , system of equations , cauchy-schwarz inequality

Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$

2016 Czech-Polish-Slovak Junior Match, 6

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

Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that $a + b + c = 3k + 1$, $ab + bc + ca = 3k^2 + 2k$. Slovakia

2000 Swedish Mathematical Competition, 6

Tags: algebra , system , system of equations

Solve \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\ y(x^3-y^3) = 7 \\ \end{array} \right. \]

2022 German National Olympiad, 1

Tags: system of equations , quadratic equation , parameterization , algebra

Determine all real numbers $a$ for which the system of equations \begin{align*} 3x^2+2y^2+2z^2&=a\\ 4x^2+4y^2+5z^2&=1-a \end{align*} has at least one solution $(x,y,z)$ in the real numbers.

2015 Turkmenistan National Math Olympiad, 1

Tags: algebra , system of equations

Solve : $y(x+y)^2=9 $ ; $y(x^3-y^3)=7$

2020 JBMO Shortlist, 1

Tags: system of equations , polynomial , algebra

Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2008 Bulgarian Autumn Math Competition, Problem 9.1

Tags: system of equations , algebra

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

2024 Bulgarian Autumn Math Competition, 10.1

Tags: algebra , system of equations

Find all real solutions to the system of equations: $$\begin{cases} (x^2+xy+y^2)\sqrt{x^2+y^2} = 88 \\ (x^2-xy+y^2)\sqrt{x^2+y^2} = 40 \end{cases}$$

1981 Vietnam National Olympiad, 1

Tags: system of equations , algebra

Solve the system of equations \[x^2 + y^2 + z^2 + t^2 = 50;\] \[x^2 - y^2 + z^2 - t^2 = -24;\] \[xy = zt;\] \[x - y + z - t = 0.\]

2001 Putnam, 2

Tags: algebra , system of equations

Find all pairs of real numbers $(x,y)$ satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

1969 IMO Longlists, 41

Tags: linear recurrence , algebra , recurrence relation , system of equations

$(MON 2)$ Given reals $x_0, x_1, \alpha, \beta$, find an expression for the solution of the system \[x_{n+2} -\alpha x_{n+1} -\beta x_n = 0, \qquad n= 0, 1, 2, \ldots\]

2009 JBMO Shortlist, 5

Tags: number theory , system of equations

Show that there are infinitely many positive integers $c$, such that the following equations both have solutions in positive integers: $(x^2 - c)(y^2 -c) = z^2 -c$ and $(x^2 + c)(y^2 - c) = z^2 - c$.

1967 IMO Longlists, 46

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

If $x,y,z$ are real numbers satisfying relations \[x+y+z = 1 \quad \textrm{and} \quad \arctan x + \arctan y + \arctan z = \frac{\pi}{4},\] prove that $x^{2n+1} + y^{2n+1} + z^{2n+1} = 1$ holds for all positive integers $n$.

2011 Romanian Master of 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]

1964 Poland - Second Round, 4

Tags: algebra , system of equations , system

Find the real numbers $ x, y, z $ satisfying the system of equations $$(z - x)(x - y) = a $$ $$(x - y)(y - z) = b$$ $$(y - z)(z - x) = c$$ where $ a, b, c $ are given real numbers.

2021 AMC 10 Fall, 14

Tags: algebra , system of equations , absolute value

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9\\ (|x|+|y|-4)^2&=1\\ \end{align*} $\textbf{(A)}\: 1\qquad\textbf{(B)} \: 2\qquad\textbf{(C)} \: 3\qquad\textbf{(D)} \: 5\qquad\textbf{(E)} \: 7$

2008 Hong Kong TST, 2

Tags: modular arithmetic , system of equations , algebra

Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]

2015 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , system of equations

For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$ $$y^3+z^3=3z+3x+4$$ $$x^3+z^3=3x+3y+4$$

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.

2010 Albania National Olympiad, 4

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

The sequence of Fibonnaci's numbers if defined from the two first digits $f_1=f_2=1$ and the formula $f_{n+2}=f_{n+1}+f_n$, $\forall n \in N$. [b](a)[/b] Prove that $f_{2010} $ is divisible by $10$. [b](b)[/b] Is $f_{1005}$ divisible by $4$? Albanian National Mathematical Olympiad 2010---12 GRADE Question 4.

1994 IMO Shortlist, 2

Tags: linear algebra , matrix , algebra , system of equations

In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.

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*}

2002 Regional Competition For Advanced Students, 2

Tags: system of equations , algebra

Solve the following system of equations over the real numbers: $2x_1 = x_5 ^2 - 23$ $4x_2 = x_1 ^2 + 7$ $6x_3 = x_2 ^2 + 14$ $8x_4 = x_3 ^2 + 23$ $10x_5 = x_4 ^2 + 34$

2012 Purple Comet Problems, 20

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

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.