Found problems: 744
2016 Japan Mathematical Olympiad Preliminary, 7
Let $a, b, c, d$ be real numbers satisfying the system of equation
$\[(a+b)(c+d)=2 \\
(a+c)(b+d)=3 \\
(a+d)(b+c)=4\]$
Find the minimum value of $a^2+b^2+c^2+d^2$.
2010 Saudi Arabia Pre-TST, 4.1
Find all triples $(a, b, c)$ of positive integers for which $$\begin{cases} a + bc=2010 \\ b + ca = 250\end{cases}$$
1983 Spain Mathematical Olympiad, 6
In a cafeteria, a glass of lemonade, three sandwiches and seven biscuits have cost $1$ shilling and $2$ pence, and a glass of lemonade, four sandwiches and $10$ biscuits they are worth $1$ shilling and $5$ pence. Find the price of:
a) a glass of lemonade, a sandwich and a cake;
b) two glasses of lemonade, three sandwiches and five biscuits.
($1$ shilling = $12$ pence).
2007 South africa National Olympiad, 2
Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.
2024 Belarus Team Selection Test, 4.1
Six integers $a,b,c,d,e,f$ satisfy:
$\begin{cases}
ace+3ebd-3bcf+3adf=5 \\
bce+acf-ade+3bdf=2
\end{cases}$
Find all possible values of $abcde$
[i]D. Bazyleu[/i]
2016 Indonesia TST, 2
Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations:
\[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]
2002 Denmark MO - Mohr Contest, 3
Two positive integers have the sum $2002$. Can $2002$ divide their product?
2018 Polish Junior MO Second Round, 1
Do positive reals $a, b, c, x$ such that $a^2+ b^2 = c^2$ and $(a + x)^2+ (b +x)^2 = (c + x)^2$ exist?
2018 German National Olympiad, 1
Find all real numbers $x,y,z$ satisfying the following system of equations:
\begin{align*}
xy+z&=-30\\
yz+x &= 30\\
zx+y &=-18
\end{align*}
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]
2024 Francophone Mathematical Olympiad, 1
Find the largest integer $k$ with the following property: Whenever real numbers $x_1,x_2,\dots,x_{2024}$ satisfy
\[x_1^2=(x_1+x_2)^2=\dots=(x_1+x_2+\dots+x_{2024})^2,\]
at least $k$ of them are equal.
2013 Dutch IMO TST, 1
Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\
bc + d + a = 5 \\
cd + a + b = 2 \\
da + b + c = 6 \end{cases}$
2018 Polish Junior MO Finals, 4
Real numbers $a, b, c$ are not equal $0$ and are solution of the system:
$\begin{cases} a^2 + a = b^2 \\ b^2 + b = c^2 \\ c^2 +c = a^2 \end{cases}$
Prove that $(a - b)(b - c)(c - a) = 1$.
1946 Moscow Mathematical Olympiad, 109
Solve the system of equations: $\begin{cases}
x_1 + x_2 + x_3 = 6 \\
x_2 + x_3 + x_4 = 9 \\
x_3 + x_4 + x_5 = 3 \\
x_4 + x_5 + x_6 = -3 \\
x_5 + x_6 + x_7 = -9 \\
x_6 + x_7 + x_8 = -6 \\
x_7 + x_8 + x_1 = -2 \\
x_8 + x_1 + x_2 = 2 \end{cases}$
2020 Junior Balkаn MO, 1
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2006 AMC 12/AHSME, 12
The parabola $ y \equal{} ax^2 \plus{} bx \plus{} c$ has vertex $ (p,p)$ and $ y$-intercept $ (0, \minus{} p)$, where $ p\neq 0$. What is $ b$?
$ \textbf{(A) } \minus{} p \qquad \textbf{(B) } 0 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } p$
1998 Czech And Slovak Olympiad IIIA, 6
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.
1954 Moscow Mathematical Olympiad, 266
Find all solutions of the system consisting of $3$ equations:
$x \left(1 - \frac{1}{2^n}\right) +y \left(1 - \frac{1}{2^{n+1}}\right) +z \left(1 - \frac{1}{2^{n+2}}\right) = 0$ for $n = 1, 2, 3$.
1991 All Soviet Union Mathematical Olympiad, 535
Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$
1961 AMC 12/AHSME, 26
For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:
${{ \textbf{(A)}\ -1221 \qquad\textbf{(B)}\ -21.5 \qquad\textbf{(C)}\ -20.5 \qquad\textbf{(D)}\ 3 }\qquad\textbf{(E)}\ 3.5 } $
1950 Poland - Second Round, 1
Solve the system of equations
$$\begin{cases} x^2+x+y=8\\
y^2+2xy+z=168\\
z^2+2yz+2xz=12480 \end{cases}$$
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 $abcde=1$.
1991 India National Olympiad, 7
Solve the following system for real $x,y,z$
\[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]
II Soros Olympiad 1995 - 96 (Russia), 11.3
Solve the system of equations
$$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$
2019 Stars of Mathematics, 1
Let $m$ be a positive integer and $n=m^2+1$. Determine all real numbers $x_1,x_2,\dotsc ,x_n$ satisfying
$$x_i=1+\frac{2mx_i^2}{x_1^2+x_2^2+\cdots +x_n^2}\quad \text{for all }i=1,2,\dotsc ,n.$$