Found problems: 744
2011 Middle European Mathematical Olympiad, 8
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
2013 BMT Spring, 1
Find the value of $a$ satisfying
\begin{align*}
a+b&=3\\
b+c&=11\\
c+a&=61
\end{align*}
2006 Federal Competition For Advanced Students, Part 2, 3
Let $ A$ be an integer not equal to $ 0$. Solve the following system of equations in $ \mathbb{Z}^3$.
$ x \plus{} y^2 \plus{} z^3 \equal{} A$
$ \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A}$
$ xy^2z^3 \equal{} A^2$
2013 Estonia Team Selection Test, 1
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
2016 JBMO TST - Turkey, 1
Find all pairs $(x, y)$ of real numbers satisfying the equations
\begin{align*} x^2+y&=xy^2 \\
2x^2y+y^2&=x+y+3xy.
\end{align*}
2004 Thailand Mathematical Olympiad, 2
Let $a$ and $b$ be real numbers such that $$\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt2.\end{cases}$$ What is the value of $a^4 + b^4$ ?
1999 AMC 12/AHSME, 17
Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$?
$ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad
\textbf{(B)}\ x \plus{} 80 \qquad
\textbf{(C)}\ \minus{}x \plus{} 118 \qquad
\textbf{(D)}\ x \plus{} 118 \qquad
\textbf{(E)}\ 0$
2013 Kosovo National Mathematical Olympiad, 3
Prove that solution of equation $y=x^2+ax+b$ and $x=y^2+cy+d$ it belong a circle.
2005 Irish Math Olympiad, 2
Let $ D,E$ and $ F$ be points on the sides $ BC,CA$ and $ AB$ respectively of a triangle $ ABC$, distinct from the vertices, such that $ AD,BE$ and $ CF$ meet at a point $ G$. Prove that if the angles $ AGE,CGD,BGF$ have equal area, then $ G$ is the centroid of $ \triangle ABC$.
1986 Swedish Mathematical Competition, 4
Prove that $x = y = z = 1$ is the only positive solution of the system \[\left\{ \begin{array}{l}
x+y^2 +z^3 = 3\\
y+z^2 +x^3 = 3\\
z+x^2 +y^3 = 3\\
\end{array} \right.
\]
2015 Swedish Mathematical Competition, 4
Solve the system of equations $$
\left\{\begin{array}{l}
x \log x+y \log y+z \log x=0\\ \\
\dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0
\end{array} \right.
$$
2010 Saudi Arabia Pre-TST, 4.4
Find all pairs $(x, y)$ of real numbers that satisfy the system of equations
$$\begin{cases} x^4 + 2z^3 - y =\sqrt3 - \dfrac14 \\
y^4 + 2y^3 - x = - \sqrt3 - \dfrac14 \end{cases}$$
2009 Mediterranean Mathematics Olympiad, 1
Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled:
- the sum of these real numbers is at least $n$.
- the sum of their squares is at most $4n$.
- the sum of their fourth powers is at least $34n$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2003 Junior Balkan Team Selection Tests - Moldova, 6
The real numbers x and у satisfy the equations
$$\begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2 \end{cases}$$
Find the numerical value of the ratio $y/x$.
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.
2012 Middle European Mathematical Olympiad, 1
Find all triplets $ (x,y,z) $ of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]
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]
2024 Kyiv City MO Round 2, Problem 1
Solve the following system of equations in real numbers:
$$\left\{\begin{array}{l}x^2=y^2+z^2,\\x^{2023}=y^{2023}+z^{2023},\\x^{2025}=y^{2025}+z^{2025}.\end{array}\right.$$
[i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]
1979 Dutch Mathematical Olympiad, 2
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
2009 Mathcenter Contest, 4
Find the values of the real numbers $x,y,z$ that correspond to the system of equations.
$$8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})$$
$$xy + yz + zx=1$$
[i](Heir of Ramanujan)[/i]
2007 Bulgarian Autumn Math Competition, Problem 8.1
Determine all real $a$, such that the solutions to the system of equations
$\begin{cases}
\frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\
(2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a
\end{cases}$
form an interval with length $\frac{32}{225}$.
1989 Romania Team Selection Test, 2
Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$
2005 Argentina National Olympiad, 1
Let $a>b>c>d$ be positive integers satisfying $a+b+c+d=502$ and $a^2-b^2+c^2-d^2=502$ . Calculate how many possible values of $ a$ are there.
2004 Czech-Polish-Slovak Match, 4
Solve in real numbers the system of equations: \begin{align*}
\frac{1}{xy}&=\frac{x}{z}+1 \\
\frac{1}{yz}&=\frac{y}{x}+1 \\
\frac{1}{zx}&=\frac{z}{y}+1 \\
\end{align*}
1969 German National Olympiad, 4
Solve the system of equations:
$$|\log_2(x + y)| + | \log_2(x - y)| = 3$$
$$xy = 3$$