Found problems: 2529
Find all pairs $(x, y)$ of integers satisfying the following condition:
each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ .
(S Zlobin)
The triangle $ ABC$ is inscribed in a circle. The interior bisectors of the angles $ A,B$ and $ C$ meet the circle again at $ A', B'$ and $ C'$ respectively. Prove that the area of triangle $ A'B'C'$ is greater than or equal to the area of triangle $ ABC.$
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Find all triples of positive real numbers $(a, b, c)$ so that the expression
$M = \frac{(a + b)(b + c)(a + b + c)}{abc}$
gets its least value.
There are $2012$ distinct points in the plane, each of which is to be coloured using one of $n$ colours, so that the numbers of points of each colour are distinct. A set of $n$ points is said to be [i]multi-coloured [/i]if their colours are distinct. Determine $n$ that maximizes the number of multi-coloured sets.
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]
Find all sets of positive integers $\{x_1, x_2, \dots, x_{20}\}$ such that $$x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1})$$ for $i=1, 2, \dots, 20$ where $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$.
Show: For all real numbers $ a,b,c$ with $ 0<a,b,c<1$ is:
\[ \sqrt{a^2bc\plus{}ab^2c\plus{}abc^2}\plus{}\sqrt{(1\minus{}a)^2(1\minus{}b)(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)^2(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)(1\minus{}c)^2}<\sqrt{3}.\]
If it is known that $\log_2a+\log_2b\geq 6$, then the least value that can be taken on by $a+b$ is:
$\textbf{(A) }2\sqrt6\qquad
\textbf{(B) }6\qquad
\textbf{(C) }8\sqrt2\qquad
\textbf{(D) }16\qquad
\textbf{(E) }\text{none of these.}$
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
Let $f(x)=\sum_{i=0}^{n}a_ix^i$ and $g(x)=\sum_{i=0}^{n}b_ix^i$, where $a_n$,$b_n$ can be zero.
Called $f(x)\ge g(x)$ if exist $r$ such that $\forall i>r,a_i=b_i,a_r>b_r$ or $f(x)=g(x)$.
Prove that: if the leading coefficients of $f$ and $g$ are positive, then $f(f(x))+g(g(x))\ge f(g(x))+g(f(x))$
Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.
Prove that if $ a,b,$ and $ c$ are positive real numbers, then \[ a^ab^bc^c \ge (abc)^{(a\plus{}b\plus{}c)/3}.\]
Given an integer $n\geq 3$. For each $3\times3$ squares on the grid, call this $3\times3$ square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white.
Now suppose one can paint an infinite grid by white or black, so that one can select an $a\times b$ rectangle which contains at least $n^2-n$ isolated $3\times 3$ square. Find the minimum of $a+b$ that such thing can happen.
(Note that $a,b$ are positive reals, and selected $a\times b$ rectangle may have sides not parallel to grid line of the infinite grid.)
Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number$ C$ such that for any positive integer $ n $ , we have $$\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C$$
Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
Let $a,b,c \in [1, \infty)$. Prove that:
$$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$
Find the greatest real number $M$ for which
\[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \]
for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$
Let $a,b,c$ be positive real numbers, such that $(ab)^2 + (bc)^2 + (ca)^2 = 3$. Prove that
\[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \]
[i]Proposed by Florin Stanescu (wer), România[/i]
Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$
The product of the positive real numbers $x, y, z$ is 1. Show that if \[ \frac{1}{x}+\frac{1}{y} + \frac{1}{z} \geq x+y+z \]then \[ \frac{1}{x^{k}}+\frac{1}{y^{k}} + \frac{1}{z^{k}} \geq x^{k}+y^{k}+z^{k} \]
for all positive integers $k$.
Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum
$$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum
Let $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.