Found problems: 744
2004 Germany Team Selection Test, 1
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
1984 All Soviet Union Mathematical Olympiad, 382
Positive $x,y,z$ satisfy a system: $\begin{cases} x^2 + xy + y^2/3= 25 \\
y^2/ 3 + z^2 = 9 \\
z^2 + zx + x^2 = 16 \end{cases}$
Find the value of expression $xy + 2yz + 3zx$.
2007 Pan African, 1
Solve the following system of equations for real $x,y$ and $z$:
\begin{eqnarray*}
x &=& \sqrt{2y+3}\\
y &=& \sqrt{2z+3}\\
z &=& \sqrt{2x+3}.
\end{eqnarray*}
2013 Hanoi Open Mathematics Competitions, 13
Solve the system of equations $\begin{cases} \frac{1}{x}+\frac{1}{y}=\frac{1}{6} \\
\frac{3}{x}+\frac{2}{y}=\frac{5}{6} \end{cases}$
2021 Bosnia and Herzegovina Junior BMO TST, 1
Determine all real numbers $a, b, c, d$ for which
$$ab + c + d = 3$$
$$bc + d + a = 5$$
$$cd + a + b = 2$$
$$da + b + c = 6$$
2017 India PRMO, 2
Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt{b} = 183, a\sqrt{b} + b\sqrt{a} = 182$. Find $\frac95 (a + b)$.
1986 IMO Longlists, 52
Solve the system of equations
\[\tan x_1 +\cot x_1=3 \tan x_2,\]\[\tan x_2 +\cot x_2=3 \tan x_3,\]\[\vdots\]\[\tan x_n +\cot x_n=3 \tan x_1\]
2024 German National Olympiad, 1
The five real numbers $v,w,x,y,s$ satisfy the system of equations
\begin{align*}
v&=wx+ys,\\
v^2&=w^2x+y^2s,\\
v^3&=w^3x+y^3s.
\end{align*}
Show that at least two of them are equal.
2004 India IMO Training Camp, 3
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
1988 IMO Longlists, 62
Let $x = p, y = q, z = r, w = s$ be the unique solution of the system of linear equations \[ x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. \] Express the solutions of the following system in terms of $p,q,r$ and $s:$ \[ x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. \] Assume the uniquness of the solution.
1973 AMC 12/AHSME, 24
The check for a luncheon of 3 sandwiches, 7 cups of coffee and one piece of pie came to $ \$3.15$. The check for a luncheon consisting of 4 sandwiches, 10 cups of coffee and one piece of pie came to $ \$4.20$ at the same place. The cost of a luncheon consisting of one sandwich, one cup of coffee, and one piece of pie at the same place will come to
$ \textbf{(A)}\ \$1.70 \qquad
\textbf{(B)}\ \$1.65 \qquad
\textbf{(C)}\ \$1.20 \qquad
\textbf{(D)}\ \$1.05 \qquad
\textbf{(E)}\ \$0.95$
1986 IMO, 1
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
1969 Poland - Second Round, 1
Prove that if the real numbers $ a, b, c, d $ satisfy the equations
$$
\; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},$$
then
$$a^2 + ac + c^2 = b^2 + bd + d^2.$$
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]
1993 IMO Shortlist, 4
Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$:
$\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$
2018 Polish Junior MO First Round, 1
Numbers $a, b, c$ are such that $3a + 4b = 3c$ and $4a - 3b = 4c$. Show that $a^2 + b^2 = c^2$.
1967 IMO Shortlist, 5
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 $n$.
2003 Swedish Mathematical Competition, 1
If $x, y, z, w$ are nonnegative real numbers satisfying \[\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\
\end{array} \right.
\] find the smallest possible value of $x$ and the values of $y, z, w$ corresponding to it.
1999 Switzerland Team Selection Test, 4
Find all real solutions $(x,y,z)$ of the system $$\begin{cases}\dfrac{4x^2}{1+4x^2}= y\\ \\\dfrac{4y^2}{1+4y^2}= z\\
\\ \dfrac{4z^2}{1+4z^2}= x \end{cases}$$
2011 Hanoi Open Mathematics Competitions, 7
Find all pairs $(x, y)$ of real numbers satisfying the system :
$\begin{cases} x + y = 3 \\
x^4 - y^4 = 8x - y \end{cases}$
2000 German National Olympiad, 1
For each real parameter $a$, find the number of real solutions to the system
$$\begin{cases} |x|+|y| = 1 , \\ x^2 +y^2 = a \end{cases}$$
2023 Brazil National Olympiad, 4
Let $x, y, z$ be three real distinct numbers such that
$$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.
2008 JBMO Shortlist, 5
Find all triples $(x, y, z)$ of real positive numbers, which satisfy the system $\begin{cases} \frac{1}{x}+\frac{4}{y}+\frac{9}{z}=3 \\ x + y + z \le 12 \end{cases}$
1979 IMO, 2
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}.$
1976 Vietnam National Olympiad, 1
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.