Found problems: 744
1971 IMO Longlists, 30
Prove that the system of equations
\[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \]
$a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?
1965 IMO Shortlist, 4
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.
2020 Chile National Olympiad, 4
Determine all three integers $(x, y, z)$ that are solutions of the system
$$x + y -z = 6$$
$$x^3 + y^3 -z^3 = 414$$
2006 Hanoi Open Mathematics Competitions, 3
Find the number of different positive integer triples $(x, y,z)$ satisfying the equations
$x^2 + y -z = 100$ and $x + y^2 - z = 124$:
1974 IMO Shortlist, 1
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).
1967 Putnam, A6
Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations
$$ a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \;\; \text{and}\;\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0 $$
for which no $x_i$ is zero. Each such solution generates a $4$-tuple of plus and minus signs (by considering the sign of $x_i$).
[list=a]
[*] Determine, with proof, the maximum number of distinct $4$-tuples possible.
[*] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$-tuples is attained.
2007 Mathematics for Its Sake, 2
For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones.
[i]Mugur Acu[/i]
V Soros Olympiad 1998 - 99 (Russia), 9.3
Solve the system of equations:
$$\frac{x-1}{xy-3}=\frac{y-2}{xy-4}=\frac{3-x-y}{7-x^2-y^2}$$
2006 China Second Round Olympiad, 3
Solve the system of equations in real numbers:
\[ \begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases} \]
2022 Poland - Second Round, 1
Find all real quadruples $(a,b,c,d)$ satisfying the system of equations
$$
\left\{ \begin{array}{ll}
ab+cd = 6 \\
ac + bd = 3 \\
ad + bc = 2 \\
a + b + c + d = 6.
\end{array} \right.
$$
1998 Akdeniz University MO, 5
Solve the equation system for real numbers:
$$x_1+x_2=x_3^2$$
$$x_2+x_3=x_4^2$$
$$x_3+x_4=x_1^2$$
$$x_4+x_1=x_2^2$$
2018 Polish Junior MO Second Round, 3
Determine all trios of integers $(x, y, z)$ which are solution of system of equations
$\begin{cases} x - yz = 1 \\ xz + y = 2 \end{cases}$
2001 AIME Problems, 11
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N+1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1=y_2,$ $x_2=y_1,$ $x_3=y_4,$ $x_4=y_5,$ and $x_5=y_3.$ Find the smallest possible value of $N.$
2002 Denmark MO - Mohr Contest, 3
Two positive integers have the sum $2002$. Can $2002$ divide their product?
2015 Czech and Slovak Olympiad III A, 4
Find all real triples $(a,b,c)$, for which $$a(b^2+c)=c(c+ab)$$ $$b(c^2+a)=a(a+bc)$$ $$c(a^2+b)=b(b+ca).$$
2018 Dutch IMO TST, 1
(a) If $c(a^3+b^3) = a(b^3+c^3) = b(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
(b) If $a(a^3+b^3) = b(b^3+c^3) = c(c^3+a^3)$ with $a, b, c$ positive real numbers,
does $a = b = c$ necessarily hold?
2024 AMC 12/AHSME, 17
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
2022 South Africa National Olympiad, 2
Find all pairs of real numbers $x$ and $y$ which satisfy the following equations:
\begin{align*}
x^2 + y^2 - 48x - 29y + 714 & = 0 \\
2xy - 29x - 48y + 756 & = 0
\end{align*}
2017 VJIMC, 3
Let $n \ge 2$ be an integer. Consider the system of equations
\begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align}
1. Prove that $(1)$ has infinitely many real solutions $(x_1,\dotsc,x_n)$ such that the numbers $x_1,\dotsc,x_n$ are distinct.
2. Prove that every solution of $(1)$, such that the numbers $x_1,\dotsc,x_n$ are not all equal, satisfies $\vert x_1x_2\cdots x_n\vert=2^{n/2}$.
1978 Czech and Slovak Olympiad III A, 3
Let $\alpha,\beta,\gamma$ be angles of a triangle. Determine all real triplets $x,y,z$ satisfying the system
\begin{align*}
x\cos\beta+\frac1z\cos\alpha &=1, \\
y\cos\gamma+\frac1x\cos\beta &=1, \\
z\cos\alpha+\frac1y\cos\gamma &=1.
\end{align*}
2019 Dutch IMO TST, 2
Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and
$\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$
2017 Regional Olympiad of Mexico Northeast, 6
Find all triples of real numbers $(a, b, c)$ that satisfy the system of equations $$\begin{cases} b^2 = 4a(\sqrt{c} - 1) \\ c^2 = 4b (\sqrt{a} - 1) \\ a^2 = 4c(\sqrt{b} - 1) \end{cases}$$
1974 IMO, 1
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).
2020 HK IMO Preliminary Selection Contest, 17
How many positive integer solutions does the following system of equations have?
$$\begin{cases}\sqrt{2020}(\sqrt{a}+\sqrt{b})=\sqrt{(c+2020)(d+2020)}\\\sqrt{2020}(\sqrt{b}+\sqrt{c})=\sqrt{(d+2020)(a+2020)}\\\sqrt{2020}(\sqrt{c}+\sqrt{d})=\sqrt{(a+2020)(b+2020)}\\\sqrt{2020}(\sqrt{d}+\sqrt{a})=\sqrt{(b+2020)(c+2020)}\\
\end{cases}$$
2025 Kosovo National Mathematical Olympiad`, P1
Find all real numbers $a$, $b$ and $c$ that satisfy the following system of equations:
$$\begin{cases}
ab-c = 3 \\
a+bc = 4 \\
a^2+c^2 = 5\end{cases}$$