Found problems: 744
1967 IMO Shortlist, 1
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
2007 Swedish Mathematical Competition, 1
Solve the following system
\[
\left\{ \begin{array}{l}
xyzu-x^3=9 \\
x+yz=\dfrac{3}{2}u \\
\end{array} \right.
\]
in positive integers $x$, $y$, $z$ and $u$.
2019 Bosnia and Herzegovina EGMO TST, 1
Let $x_1,x_2, ..., x_n$ be non-negative real numbers. Solve the system of equations:
$$x_k+x_{k+1}=x^2_{k+2}\,\,,\,\,\, (k =1,2,...,n),$$
where $x_{n+1} = x_1$, $x_{n+2} = x_2$.
2005 German National Olympiad, 1
Find all pairs (x; y) of real numbers satisfying the system of equations
$x^3 + 1 - xy^2 - y^2 = 0$;
$y^3 - 1 - x^2y + x^2 = 0$.
Darij
2022 Bulgarian Spring Math Competition, Problem 10.4
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy
\[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]
2016 Middle European Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
2003 German National Olympiad, 1
Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$
2018 Czech-Polish-Slovak Junior Match, 1
Are there four real numbers $a, b, c, d$ for every three positive real numbers $x, y, z$ with the property $ad + bc = x$, $ac + bd = y$, $ab + cd = z$ and one of the numbers $a, b, c, d$ is equal to the sum of the other three?
2012 European Mathematical Cup, 3
Are there positive real numbers $x$, $y$ and $z$ such that
$ x^4 + y^4 + z^4 = 13\text{,} $
$ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $
$ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $
[i]Proposed by Matko Ljulj.[/i]
2019 CMI B.Sc. Entrance Exam, 5
Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$
1976 IMO, 2
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
2025 Ukraine National Mathematical Olympiad, 9.1
Solve the system of equations in reals:
\[
\begin{cases}
y = x^2 + 2x \\
z = y^2 + 2y \\
x = z^2 + 2z
\end{cases}
\]
[i]Proposed by Mykhailo Shtandenko[/i]
2023 German National Olympiad, 4
Determine all triples $(a,b,c)$ of real numbers with
\[a+\frac{4}{b}=b+\frac{4}{c}=c+\frac{4}{a}.\]
1992 All Soviet Union Mathematical Olympiad, 564
Find all real $x, y$ such that $$\begin{cases}(1 + x)(1 + x^2)(1 + x^4) = 1+ y^7 \\
(1 + y)(1 + y^2)(1 + y^4) = 1+ x^7 \end{cases}$$
1999 Vietnam National Olympiad, 1
Solve the system of equations:
$ (1\plus{}4^{2x\minus{}y}).5^{1\minus{}2x\plus{}y}\equal{}1\plus{}2^{2x\minus{}y\plus{}1}$
$ y^3\plus{}4x\plus{}ln(y^2\plus{}2x)\plus{}1\equal{}0$
1984 IMO Shortlist, 9
Let $a, b, c$ be positive numbers with $\sqrt a +\sqrt b +\sqrt c = \frac{\sqrt 3}{2}$. Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1,\] \[\sqrt{z-b}+\sqrt{x-b}=1,\] \[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x, y, z)$ in real numbers.
1979 IMO Longlists, 44
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
2014 Hanoi Open Mathematics Competitions, 9
Solve the system $\begin {cases} 16x^3 + 4x = 16y + 5 \\
16y^3 + 4y = 16x + 5 \end{cases}$
2023 Bulgaria JBMO TST, 1
Determine all triples $(x,y,z)$ of real numbers such that $x^4 + y^3z = zx$, $y^4 + z^3x = xy$ and $z^4 + x^3y = yz$.
2010 Morocco TST, 1
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?
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]
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]
2017 CMIMC Algebra, 7
Let $a$, $b$, and $c$ be complex numbers satisfying the system of equations \begin{align*}\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}&=9,\\\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}&=32,\\\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}&=122.\end{align*} Find $abc$.
2005 Estonia National Olympiad, 1
Real numbers $x$ and $y$ satisfy the system of equalities
$$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$
Prove that $\cos 2x = \cos 2y$.
1990 Austrian-Polish Competition, 4
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$