Found problems: 744
1984 IMO Longlists, 17
Find all solutions of the following system of $n$ equations in $n$ variables:
\[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\]
where $a$ is a given number.
2008 JBMO Shortlist, 3
Let the real parameter $p$ be such that the system $\begin{cases} p(x^2 - y^2) = (p^2- 1)xy \\ |x - 1|+ |y| = 1 \end{cases}$ has at least three different real solutions. Find $p$ and solve the system for that $p$.
1999 Moldova Team Selection Test, 2
Let $a,b,c$ be positive numbers. Prove that a triangle with sides $a,b,c$ exists if and only if the system of equations
$$\begin{cases}\dfrac{y}{z}+\dfrac{z}{y}=\dfrac{a}{x} \\ \\ \dfrac{z}{x}+\dfrac{x}{z}=\dfrac{b}{y} \\ \\ \dfrac{x}{y}+\dfrac{y}{x}=\dfrac{c}{z}\end{cases}$$ has a real solution.
2023 Greece Junior Math Olympiad, 1
Solve in real numbers the system:
$$\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}$$
2005 Austrian-Polish Competition, 7
For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$:
$$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$
where
$$S=x_1^2+x_2^2+\dots+x_n^2.$$
2010 Mediterranean Mathematics Olympiad, 1
Real numbers $a,b,c,d$ are given. Solve the system of equations (unknowns $x,y,z,u)$\[
x^{2}-yz-zu-yu=a\]
\[
y^{2}-zu-ux-xz=b\]
\[
z^{2}-ux-xy-yu=c\]
\[
u^{2}-xy-yz-zx=d\]
2012 Argentina National Olympiad, 1
Determine if there are triplets ($x,y,z)$ of real numbers such that
$$\begin{cases} x+y+z=7 \\ xy+yz+zx=11\end{cases}$$
If the answer is affirmative, find the minimum and maximum values of $z$ in such a triplet.
2023 Poland - Second Round, 4
Given pairwise different real numbers $a,b,c,d,e$ such that
$$
\left\{ \begin{array}{ll}
ab + b = ac + a, \\
bc + c = bd + b, \\
cd + d = ce + c, \\
de + e = da + d.
\end{array} \right.
$$
Prove that $abcde=1$.
2022 Puerto Rico Team Selection Test, 1
Find all triples $(a, b, c)$ of positive integers such that:
$$a + b + c = 24$$
$$a^2 + b^2 + c^2 = 210$$
$$abc = 440$$
2012 India Regional Mathematical Olympiad, 1
Find with proof all nonzero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.
2002 Abels Math Contest (Norwegian MO), 2c
If $a$ and $b$ are real numbers such that $$\begin{cases} a^3-3ab^2 = 8 \\ b^3-3a^2b = 11 \end{cases}$$ then what is $a^2+b^2$?
KoMaL A Problems 2020/2021, A. 788
Solve the following system of equations:
$$x+\frac{1}{x^3}=2y,\quad y+\frac{1}{y^3}=2z,\quad z+\frac{1}{z^3}=2w,\quad w+\frac{1}{w^3}=2x.$$
1980 AMC 12/AHSME, 14
If the function $f$ is defined by
\[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is
$\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$
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 $(x , y , z )$ :
$\begin{cases}x + y = a - 1 \\
x(y + 1) - z^2 = b \end{cases}$
1983 Swedish Mathematical Competition, 6
Show that the only real solution to
\[\left\{ \begin{array}{l}
x(x+y)^2 = 9 \\
x(y^3 - x^3) = 7 \\
\end{array} \right.
\]
is $x = 1$, $y = 2$.
II Soros Olympiad 1995 - 96 (Russia), 10.4
Solve the system of equations
$$\begin{cases} x^2+ [y]=10
\\ y^2+[x]=13
\end{cases}$$
($[x]$ is the integer part of $x$, $[x]$ is equal to the largest integer not exceeding $x$. For example, $[3,33] = 3$, $[2] = 2$, $[- 3.01] = -4$).
2010 Dutch IMO TST, 5
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
1953 Poland - Second Round, 4
Solve the system of equations $$ \qquad<br /> \begin{array}{c}<br /> x_1x_2 = 1\\<br /> x_2x_3 = 2\\<br /> x_3x_4 = 3\\<br /> \ldots\\<br /> x_nx_1 = n<br /> \end{array}$$
1935 Moscow Mathematical Olympiad, 017
Solve the system $\begin{cases} x^3 - y^3 = 26 \\
x^2y - xy^2 = 6
\end{cases}$ in $C$
[hide=other version]solved below
Solve the system $\begin{cases} x^3 - y^3 = 2b \\
x^2y - xy^2 = b
\end{cases}$[/hide]
2015 Junior Balkan Team Selection Tests - Romania, 2
Find all the triplets of real numbers $(x , y , z)$ such that :
$y=\frac{x^3+12x}{3x^2+4}$ , $z=\frac{y^3+12y}{3y^2+4}$ , $x=\frac{z^3+12z}{3z^2+4}$
2002 Regional Competition For Advanced Students, 2
Solve the following system of equations over the real numbers:
$2x_1 = x_5 ^2 - 23$
$4x_2 = x_1 ^2 + 7$
$6x_3 = x_2 ^2 + 14$
$8x_4 = x_3 ^2 + 23$
$10x_5 = x_4 ^2 + 34$
1985 ITAMO, 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]
2000 German National Olympiad, 4
Find all nonnegative solutions $(x,y,z)$ to the system
$$\begin{cases} \sqrt{x+y}+\sqrt{z} = 7 \\
\sqrt{x+z}+\sqrt{y} = 7 \\
\sqrt{y+z}+\sqrt{x} = 5 \end{cases}$$
1981 Swedish Mathematical Competition, 2
Does
\[\left\{ \begin{array}{l}
x^y = z \\
y^z = x \\
z^x = y \\
\end{array} \right.
\]
have any solutions in positive reals apart from $x = y = z= 1$?
2001 Grosman Memorial Mathematical Olympiad, 1
Find all real solutions of the system
$$\begin{cases} x_1 +x_2 +...+x_{2000} = 2000 \\ x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases}$$