Found problems: 744
2016 Baltic Way, 9
Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations:
$$\begin{cases} a^3 + c^3 = 2 \\ a^2b + c^2d = 0 \\ b^3 + d^3 = 1 \\ ab^2 + cd^2 = -6.\end{cases}$$
1996 AIME Problems, 1
In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x.$
[asy]
size(100);defaultpen(linewidth(0.7));
int i;
for(i=0; i<4; i=i+1) {
draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6));
}
label("$x$", (1,5));
label("$1$", (1,3));
label("$19$", (3,5));
label("$96$", (5,5));[/asy]
1979 All Soviet Union Mathematical Olympiad, 276
Find $x$ and $y$ ($a$ and $b$ parameters):
$$\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}$$
V Soros Olympiad 1998 - 99 (Russia), 9.3
Solve the system of equations:
$$\begin{cases} x + [y] + \{z\}=3.9 \\
y + [z] + \{x\}= 3.5 \\
z + [x] + \{y\}= 2.
\end{cases}$$
2023-IMOC, N6
Let $S(b)$ be the number of nonuples of positive integers $(a_1, a_2, \ldots , a_9)$ satisfying $3b-1=a_1+a_2+\ldots+a_9$ and $b^2+1=a_1^2+\ldots+a_9^2$. Prove that for all $\epsilon>0$, there exists $C_{\epsilon}>0$ such that $S(b)\leq C_{\epsilon}b^{3+\epsilon}$.
2024 Indonesia MO, 1
Determine all positive real solutions $(a,b)$ to the following system of equations.
\begin{align*} \sqrt{a} + \sqrt{b} &= 6 \\ \sqrt{a-5} + \sqrt{b-5} &= 4 \end{align*}
2005 Kazakhstan National Olympiad, 1
Does there exist a solution in real numbers of the system of equations
\[\left\{
\begin{array}{rcl}
(x - y)(z - t)(z - x)(z - t)^2 = A, \\
(y - z)(t - x)(t - y)(x - z)^2 = B,\\
(x - z)(y - t)(z - t)(y - z)^2 = C,\\
\end{array}
\right.\]
when
a) $A=2, B=8, C=6;$
b) $A=2, B=6, C=8.$?
2010 Czech-Polish-Slovak Match, 1
Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations
\[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]
2006 Czech and Slovak Olympiad III A, 6
Find all real solutions $(x,y,z)$ of the system of equations:
\[
\begin{cases}
\tan ^2x+2\cot^22y=1 \\
\tan^2y+2\cot^22z=1 \\
\tan^2z+2\cot^22x=1 \\
\end{cases}
\]
2010 Contests, 3
Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations:
\[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\]
\[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]
2003 AMC 8, 4
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 7$
1988 Tournament Of Towns, (170) 3
Find all real solutions of the system of equations
$$\begin{cases} (x_3 + x_4 + x_5)^5 = 3x_1 \\
(x_4 + x_5 + x_1)^5 = 3x_2\\
(x_5 + x _1 + x_2)^5 = 3x_3\\
(x_1 + x_2 + x_3)^5 = 3x_4\\
(x_2 + x_3 + x_4)^5 = 3x_5 \end{cases}$$
(L. Tumescu , Romania)
2005 Junior Tuymaada Olympiad, 3
Tram ticket costs $1$ Tug ($=100$ tugriks). $20$ passengers have only coins in denominations of $2$ and $5$ tugriks, while the conductor has nothing at all. It turned out that all passengers were able to pay the fare and get change. What is the smallest total number of passengers that the tram could have?
2008 Mathcenter Contest, 1
Given $x,y,z\in \mathbb{R} ^+$ , that are the solutions to the system of equations :
$$x^2+xy+y^2=57$$
$$y^2+yz+z^2=84$$
$$z^2+zx+x^2=111$$
What is the value of $xy+3yz+5zx$?
[i](maphybich)[/i]
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.
2019 German National Olympiad, 6
Suppose that real numbers $x,y$ and $z$ satisfy the following equations:
\begin{align*}
x+\frac{y}{z} &=2,\\
y+\frac{z}{x} &=2,\\
z+\frac{x}{y} &=2.
\end{align*}
Show that $s=x+y+z$ must be equal to $3$ or $7$.
[i]Note:[/i] It is not required to show the existence of such numbers $x,y,z$.
2018 JBMO Shortlist, A4
Let $k > 1, n > 2018$ be positive integers, and let $n$ be odd. The nonzero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and satisfy $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$
Find:
a) the product $x_1 x_2 \ldots x_n$ as a function of $k$ and $n$
b) the least value of $k$, such that there exist $n,x_1,x_2,\ldots,x_n$ satisfying the given conditions.
1994 Tuymaada Olympiad, 7
Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$
2004 Czech and Slovak Olympiad III A, 2
Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\]
1970 IMO Longlists, 4
Solve the system of equations for variables $x,y$, where $\{a,b\}\in\mathbb{R}$ are constants and $a\neq 0$.
\[x^2 + xy = a^2 + ab\] \[y^2 + xy = a^2 - ab\]
2010 Saint Petersburg Mathematical Olympiad, 1
Solve in positives $$x^y=z,y^z=x,z^x=y$$
2019 Durer Math Competition Finals, 13
Let $k > 1$ be a positive integer and $n \ge 2019$ be an odd positive integer. The non-zero rational numbers $x_1, x_2,..., x_n$ are not all equal, and satisfy the following chain of equalities:
$$x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}.$$
What is the smallest possible value of $k$?
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 "?"
2022 Latvia Baltic Way TST, P1
Find all triplets of positive real numbers $(x,y,z)$ that satisfy the following system of equations:
$$ \begin{cases}
x+y^2+z^3=3\\
y+z^2+x^3=3\\
z+x^2+y^3=3.
\end{cases}$$
2013 Dutch Mathematical Olympiad, 2
Find all triples $(x, y, z)$ of real numbers satisfying:
$x + y - z = -1$ , $x^2 - y^2 + z^2 = 1$ and $- x^3 + y^3 + z^3 = -1$