Found problems: 451
1969 IMO Shortlist, 17
$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$
1982 IMO Longlists, 31
Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
1989 ITAMO, 1
Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.
1968 IMO Shortlist, 15
Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \]
[hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]
2015 Hanoi Open Mathematics Competitions, 8
Solve the equation $(2015x -2014)^3 = 8(x-1)^3 + (2013x -2012)^3$
2015 Finnish National High School Mathematics Comp, 1
Solve the equation $\sqrt{1+\sqrt {1+x}}=\sqrt[3]{x}$ for $x \ge 0$.
1995 IMO Shortlist, 4
Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.
2003 Poland - Second Round, 6
Each pair $(x, y)$ of nonnegative integers is assigned number $f(x, y)$ according the conditions:
$f(0, 0) = 0$;
$f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$,
$f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$ for $x, y \ge 0$.
Let $n$ be a fixed nonnegative integer and let $a$, $b$ be nonnegative integers such that $f(a, b) = n$. Decide how many numbers satisfy the equation $f(a, x) + f(b, x) = n$.
2011 Ukraine Team Selection Test, 7
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2002 Baltic Way, 1
Solve the system of simultaneous equations
\[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\]
in real numbers.
1980 All Soviet Union Mathematical Olympiad, 301
Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$.
($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)
1984 IMO Longlists, 32
Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$.
(a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$
(b) Prove that $SD + SE + SF = 2(SA + SB + SC).$
2008 India National Olympiad, 2
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.
1977 IMO Longlists, 28
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
2000 Tournament Of Towns, 4
Let $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation
$$a_1 +\dfrac{1}{a_2+\dfrac{1}{a_3+ ... \dfrac{1}{a_n+\dfrac{1}{x}} } } = x$$
for all values of $x$ for which the lefthand side of the equation makes sense.
(a) Prove that $n$ is even.
(b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist?
(M Skopenko)
1959 IMO, 2
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given
a) $A=\sqrt{2}$;
b) $A=1$;
c) $A=2$,
where only non-negative real numbers are admitted for square roots?
1961 IMO, 3
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
1959 IMO Shortlist, 2
For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given
a) $A=\sqrt{2}$;
b) $A=1$;
c) $A=2$,
where only non-negative real numbers are admitted for square roots?
1989 IMO Longlists, 8
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
2023 Myanmar IMO Training, 4
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
2005 India IMO Training Camp, 2
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
2015 Bosnia and Herzegovina Junior BMO TST, 1
Solve equation $x(x+1) = y(y+4)$ where $x$, $y$ are positive integers
2017 Junior Regional Olympiad - FBH, 4
Group of $27$ climbers shared among themself $13$ breads. Every man had $2$ breads, every woman half of a bread, and every child $\frac{1}{3}$ of a bread. How many men, women and children where there ?
1976 Dutch Mathematical Olympiad, 4
For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?
1996 Tuymaada Olympiad, 5
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$