Found problems: 451
Let $ a$ and $ b$ be non-negative integers such that $ ab \geq c^2,$ where $ c$ is an integer. Prove that there is a number $ n$ and integers $ x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n$ such that
\[ \sum^n_{i\equal{}1} x^2_i \equal{} a, \sum^n_{i\equal{}1} y^2_i \equal{} b, \text{ and } \sum^n_{i\equal{}1} x_iy_i \equal{} c.\]
Find all real numbers $p$ such that the equation $$\sqrt{x^2-5p^2}=px-1$$ has a root $x=3$. Then, solve the equation for the determined values of $p$.
Determine all triples $(a,b,c)$ of integers that simultaneously satisfy the following relations:
\begin{align*}
a^2 + a = b + c, \\
b^2 + b = a + c, \\
c^2 + c = a + b.
\end{align*}
Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.
Solve the equation $(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
Find all the real numbers $a$ and $b$ that satisfy the relation
$2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)$
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
Determine, with proof, all triples of real numbers $(x,y,z)$ satisfying the equations
$$x^3+y+z=x+y^3+z=x+y+z^3=-xyz.$$
Find all pairs $(a,b)$ of real numbers, so that $\sin(2009x)+\sin(ax)+\sin(bx)=0$ holds for any $x\in \mathbf {R}$.
We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that
$$P(x)=\sum_{}{x^{n_i}}+1$$
Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.
Find all real numbers $x$ such that $4^x-2^{x+2}+3=0$.
Find all positive integer solutions $ x, y, z$ of the equation $ 3^x \plus{} 4^y \equal{} 5^z.$
Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.
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]
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
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 $n$.
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]
Let be a natural number $ n\ge 3. $ Solve the equation $ \lfloor x/n \rfloor =\lfloor x-n \rfloor $ in $ \mathbb{R} . $
[i]Constantin Nicolau[/i]
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
Find all real solutions of the equation $x^2 + 2x \sin (xy) + 1 = 0$.
Prove:
(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$
(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$
Solve in $ \mathbb{N}^2: $
$$ x+y=\sqrt x+\sqrt y+\sqrt{xy} . $$
The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$.
Find all numbers $n$ such that the value of this expression is $13$.