Found problems: 451
Solve in the real numbers the equation $ 3^{x+1}=(x-1)(x-3). $
Solve in the set of real numbers the equation
$$ 16\{ x \}^2-8x=-1, $$
where $ \{\} $ denotes the fractional part.
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 \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Find all triples $(a, b, c)$ of real numbers such that $ab + bc + ca = 1$ and
$$a^2b + c = b^2c + a = c^2a + b.$$
Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation
\[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]
Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} =
\frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.
\]
[i]Proposed by Dusan Djukic, Serbia[/i]
Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$
Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that
$$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$
Find all real solutions of equation $x + \sqrt{2p - x^2} = 8$ with real parameter $p$.
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 $SD + SE + SF = 2(SA + SB + SC).$
Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) = 1$, then $x+y = 0$.
Find $x$ such that trigonometric
\[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\]
where $m$ is a fixed real number.
Prove that there exists a function $f : \mathbb{N} \rightarrow \mathbb{N}$ that satisfies the following
(1) $\{f(n) : n\in\mathbb{N}\}$ is a finite set; and
(2) For nonzero integers $x_1, x_2, \ldots, x_{1000}$ that satisfy $f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)$, then $x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0$.
Solve equation in real numbers $\log_{2}(4^x+4)=x+\log_{2}(2^{x+1}-3)$
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]
Find all pairs of real numbers $(x, y)$ that satisfy the equation $\frac{x + 6}{y}+\frac{13}{xy}=\frac{4-y}{x}$
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 $x_r - x_s = k.$
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. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?
Number of solutions of the equation $3^x+4^x=8^x$ in reals is
$\textbf{(A)}~0$
$\textbf{(B)}~1$
$\textbf{(C)}~2$
$\textbf{(D)}~\infty$
Find the real numbers $ m $ which have the property that the equation
$$ x^2-2mx+2m^2=25 $$
has two integer solutions.
$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. [hide="For example"] P could be $3*5$, but not $3^2*5$.[/hide] Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$.
[b]Original Statement:[/b]
Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which \[ ax^2 + 2bxy + cy^2 = n. \] Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.
Solve in the real numbers the equation $ \sqrt[5]{2^x-2^{-1}} -\sqrt[5]{2^x+2^{-1}} =-1. $
[i]Mihai Neagu[/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]
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.