Found problems: 451
Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $
Prove that the product of five consecutive positive integers cannot be the square of an integer.
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]
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions
Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$:
\begin{align*} x^2+py+q&=0,\\
y^2+px+q&=0.
\end{align*}
Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.
The sequence of real numbers $a_1,a_2,...,a_{2015}$ is such that the 2015 equations:
$a_1^3=a_1^2;a_1^3+a_2^3=(a_1+a_2 )^2;...;a_1^3+a_2^3+...+a_{2015}^3=(a_1+a_2+...+a_{2015} )^2$
are true. Prove that $a_1,a_2,…,a_{2015}$ are integers.
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $
[i]Dragoș Crișan[/i]
Find all real solutions of the equation
\[17^x+2^x=11^x+2^{3x}.\]
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.
Let $n$ be a natural number. Solve the equation
$$||\cdots|||x-1|-2|-3|-\ldots-(n-1)|-n|=0.$$
Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.
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 $P(n) = 15n^2$.
$\textbf{N1.}$ Find all nonnegative integers $a,b,c$ such that
\begin{align*} a^2+b^2+c^2-ab-bc-ca = a+b+c \end{align*}
[i]Proposed by usjl[/i]
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?
Solve the equation $x^3 - [x] = 3$.
$x\sqrt8 + \frac{1}{x\sqrt8} = \sqrt8$ has two real solutions $x_1, x_2$. The decimal expansion of $x_1$ has the digit $6$ in place $1994$. What digit does $x_2$ have in place $1994$?
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. If the two rooms have dimensions of 38 feet by 55 feet and 50 feet by 55 feet, what are the carpet dimensions?
Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$
In set of positive integers solve the equation $$x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)$$
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.