Found problems: 451
For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?
Find all triples of non-negative integers \((a, b, c)\) such that \[a^{2}+b^{2}+c^{2} = a b c+1.\]
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
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation
$$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$
has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $
[b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.
Consider the numbers $a,b, \alpha, \beta \in \mathbb{R}$ and the sets $$A=\left \{x \in \mathbb{R} : x^2+a|x|+b=0 \right \},$$ $$B=\left \{ x \in \mathbb{R} : \lfloor x \rfloor^2 + \alpha \lfloor x \rfloor + \beta = 0\right \}.$$
If $A \cap B$ has exactly three elements, prove that $a$ cannot be an integer.
Solve the equation $$x^2+y^2+z^2=686$$ where $x$, $y$ and $z$ are positive integers
Find the real numbers $ x,y,z $ that satisfy the following:
$ \text{(i)} -2\le x\le y\le z $
$ \text{(ii)} x+y+z=2/3 $
$ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $
[i]Cristinel Mortici[/i]
Solve the equation
$$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$
Find all real numbers $k $ such that $x^2 - 2 x [x] + x - k = 0$ has at least two non-negative roots.
Solve in $ \mathbb{R}^3 $ the system:
$$ \left\{ \begin{matrix} 3^x+4^x=5^y \\8^y+15^y=17^z \\ 20^z+21^z=29^x \end{matrix} \right. $$
[i]Cristinel Mortici[/i]
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]
Find the nonnegative real numbers $ a,b,c,d $ that satisfy the following system:
$$ \left\{ \begin{matrix} a^3+2abc+bcd-6&=&a \\a^2b+b^2c+abd+bd^2&=&b\\a^2b+a^2c+bc^2+cd^2&=&c\\d^3+ab^2+abc+bcd-6&=&d \end{matrix} \right. $$
Solve $ x^3-y^3=2xy+7 $ in integers.
Determine all real numbers that satisfy the equation $$(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$$
(RM Kuznec)
[b]a)[/b] Let be a nonnegative integer $ n. $ Solve in the complex numbers the equation $ z^n\cdot\Re z=\bar z^n\cdot\Im z. $
[b]b)[/b] Let be two complex numbers $ v,d $ satisfying $ v+1/v=d/\bar d +\bar d/d. $ Show that
$$ v^n+1/v^n=d^n/\bar d^n + \bar d^n/d^n, $$
for any nonnegative integer $ n. $
Let $a_1, a_2, \ldots, a_n$ be real numbers, but not all of them null. Show that the equation $$\sqrt{x+a_1}+\sqrt{x+a_2}+\ldots+\sqrt{x+a_n}=n\sqrt{x}$$ has at most one real solution.
Let $n$ be a given positive integer. Find the solution set of the equation $\sum_{k=1}^{2n} \sqrt{x^2 -2kx + k^2} =| 2nx - n - 2n^2|$
Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation
\[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\]
Show that at least two of these integers are equal to each other.
Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$
Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$
Find all rational numbers $x,y$ such that \[\left(x+y\sqrt5\right)^2=7+3\sqrt5.\]
Find all natural numbers $n$ the product of whose decimal digits is $n^2-10n-22$.
Solve in the real numbers the equation $ |\sin 3x+\cos (7\pi /2 -5x)|=2. $
Let $a,b,c $ be the lengths of the three sides of a triangle and $a,b$ be the two roots of the equation $ax^2-bx+c=0 $$ (a<b) . $ Find the value range of $ a+b-c .$
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
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]