Found problems: 744
2010 Contests, 1
Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations
\[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]
1974 IMO Longlists, 40
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).
1945 Moscow Mathematical Olympiad, 097
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?
2001 Rioplatense Mathematical Olympiad, Level 3, 1
Find all integer numbers $a, b, m$ and $n$, such that the following two equalities are verified:
$a^2+b^2=5mn$ and $m^2+n^2=5ab$
2016 Canada National Olympiad, 2
Consider the following system of $10$ equations in $10$ real variables $v_1, \ldots, v_{10}$:
\[v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).\]
Find all $10$-tuples $(v_1, v_2, \ldots , v_{10})$ that are solutions of this system.
2002 Greece JBMO TST, 1
Real numbers $x,y,a$ are such that $x+y=x^2+y^2=x^3+y^3=a$. Find all the possible values of $a$.
2001 Flanders Math Olympiad, 2
Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.
Find the "?"
2019 Dutch IMO TST, 2
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and
$\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
1979 All Soviet Union Mathematical Olympiad, 276
Find $x$ and $y$ ($a$ and $b$ parameters):
$$\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}$$
II Soros Olympiad 1995 - 96 (Russia), 10.7
Let us denote by $<a>$ the distance from $a$ to the nearest integer. (For example, $<1,2> = 0.2$, $<\sqrt3> = 2-\sqrt3$) How many solutions does the system of equations have
$$\begin{cases} <19x>=y
\\ <96y>=x
\end{cases} \,\,\, ?$$
2003 Bundeswettbewerb Mathematik, 2
Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations:
$x^3-4x^2-16x+60=y$;
$y^3-4y^2-16y+60=z$;
$z^3-4z^2-16z+60=x$.
1985 AIME Problems, 6
As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.
[asy]
size(200);
pair A=origin, B=(14,0), C=(9,12), D=foot(A, B,C), E=foot(B, A, C), F=foot(C, A, B), H=orthocenter(A, B, C);
draw(F--C--A--B--C^^A--D^^B--E);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("84", centroid(H, C, E), fontsize(9.5));
label("35", centroid(H, B, D), fontsize(9.5));
label("30", centroid(H, F, B), fontsize(9.5));
label("40", centroid(H, A, F), fontsize(9.5));[/asy]
1967 IMO Longlists, 7
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
1984 IMO Longlists, 44
Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$
Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1\]
\[\sqrt{z-b}+\sqrt{x-b}=1\]
\[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x,y,z)$ in real numbers.
It was proposed by Poland. Have fun! :lol:
1985 Tournament Of Towns, (082) T3
Find all real solutions of the system of equations $\begin{cases}
(x + y) ^3 = z \\ (y + z) ^3 = x \\ ( z+ x) ^3 = y \end{cases} $
(Based on an idea by A . Aho , J. Hop croft , J. Ullman )
2005 AIME Problems, 3
An infinite geometric series has sum $2005$. A new series, obtained by squaring each term of the original series, has $10$ times the sum of the original series. The common ratio of the original series is $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers. Find $m+n$.
III Soros Olympiad 1996 - 97 (Russia), 10.4
Solve the system of equations
$$\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}$$
2024 Israel National Olympiad (Gillis), P1
Solve the following system (over the real numbers):
\[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\
3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]
2009 Junior Balkan Team Selection Test, 1
Given are natural numbers $ a,b$ and $ n$ such that $ a^2\plus{}2nb^2$ is a complete square. Prove that the number $ a^2\plus{}nb^2$ can be written as a sum of squares of $ 2$ natural numbers.
2016 Baltic Way, 9
Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations:
$$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$
2020 Tuymaada Olympiad, 1
Does the system of equation
\begin{align*}
\begin{cases}
x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\
x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\
x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3
\end{cases}
\end{align*}
admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?
2014 Singapore Senior Math Olympiad, 2
Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations:
\[a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b.\]
2010 Indonesia TST, 1
Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations
$x^5 = 2y^3 + y - 2$
$y^5 = 2z^3 + z - 2$
$z^5 = 2x^3 + x - 2$
2012 European Mathematical Cup, 3
Are there positive real numbers $x$, $y$ and $z$ such that
$ x^4 + y^4 + z^4 = 13\text{,} $
$ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $
$ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $
[i]Proposed by Matko Ljulj.[/i]
2017 Hanoi Open Mathematics Competitions, 6
Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ :
$\begin{cases}x + y = a - 1 \\
x(y + 1) - z^2 = b \end{cases}$