Found problems: 451
1999 Bosnia and Herzegovina Team Selection Test, 1
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$ $$x^2-2cx+2ab=0$$ $$x^2-2ax+2bc=0$$ does not have real solutions
2017 EGMO, 5
Let $n\geq2$ be an integer. An $n$-tuple $(a_1,a_2,\dots,a_n)$ of not necessarily different positive integers is [i]expensive[/i] if there exists a positive integer $k$ such that $$(a_1+a_2)(a_2+a_3)\dots(a_{n-1}+a_n)(a_n+a_1)=2^{2k-1}.$$
a) Find all integers $n\geq2$ for which there exists an expensive $n$-tuple.
b) Prove that for every odd positive integer $m$ there exists an integer $n\geq2$ such that $m$ belongs to an expensive $n$-tuple.
[i]There are exactly $n$ factors in the product on the left hand side.[/i]
Taiwan TST 2015 Round 1, 1
Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]
[i]Proposed by Titu Andreescu, USA[/i]
2000 District Olympiad (Hunedoara), 1
[b]a)[/b] Solve the system
$$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$
[b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $
2011 Bosnia And Herzegovina - Regional Olympiad, 2
For positive integers $a$ and $b$ holds $a^3+4a=b^2$. Prove that $a=2t^2$ for some positive integer $t$
1966 IMO Shortlist, 48
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
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 $f(a, x) + f(b, x) = n$.
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.
1996 Greece Junior Math Olympiad, 1
Solve the equation
$(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$
2013 Dutch Mathematical Olympiad, 4
For a positive integer n the number $P(n)$ is the product of the positive divisors of $n$. For example, $P(20) = 8000$, as the positive divisors of $20$ are $1, 2, 4, 5, 10$ and $20$, whose product is $1 \cdot 2 \cdot 4 \cdot 5 \cdot 10 \cdot 20 = 8000$.
(a) Find all positive integers $n$ satisfying $P(n) = 15n$.
(b) Show that there exists no positive integer $n$ such that $P(n) = 15n^2$.
2008 India Regional Mathematical Olympiad, 2
Solve the system of equation
$$x+y+z=2;$$$$(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;$$$$x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.$$
2002 IMO Shortlist, 4
Is there a positive integer $m$ such that the equation \[ {1\over a}+{1\over b}+{1\over c}+{1\over abc}={m\over a+b+c} \] has infinitely many solutions in positive integers $a,b,c$?
1966 German National Olympiad, 5
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
1992 Nordic, 1
Determine all real numbers $x > 1, y > 1$, and $z > 1$,satisfying the equation
$x+y+z+\frac{3}{x-1}+\frac{3}{y-1}+\frac{3}{z-1}=2(\sqrt{x+2}+\sqrt{y+2}+\sqrt{z+2})$
1997 IMO Shortlist, 17
Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.
2016 Junior Regional Olympiad - FBH, 4
In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$
2011 Croatia Team Selection Test, 4
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
1985 Brazil National Olympiad, 5
$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.
2017 District Olympiad, 2
Let $ E(x,y)=\frac{x}{y} +\frac{x+1}{y+1} +\frac{x+2}{y+2} . $
[b]a)[/b] Solve in $ \mathbb{N}^2 $ the equation $ E(x,y)=3. $
[b]b)[/b] Show that there are infinitely many natural numbers $ n $ such that the equation $ E(x,y)=n $ has at least one solution in $ \mathbb{N}^2. $
2010 VTRMC, Problem 3
Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$
1990 India National Olympiad, 1
Given the equation
\[ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0\]
has four real, positive roots, prove that
(a) $ pr \minus{} 16s \geq 0$
(b) $ q^2 \minus{} 36s \geq 0$
with equality in each case holding if and only if the four roots are equal.
2016 Dutch BxMO TST, 1
For a positive integer $n$ that is not a power of two, we de fine $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.
2025 District Olympiad, P3
Determine all positive real numbers $a,b,c,d$ such that $a+b+c+d=80$ and $$a+\frac{b}{1+a}+\frac{c}{1+a+b}+\frac{d}{1+a+b+c}=8.$$
2005 Germany Team Selection Test, 1
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 $n$.
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$