Found problems: 85335
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.
Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.
[b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^2}\sum_{i=0}^n\sqrt{\binom{n+i}{2}} $
[b]b)[/b] $ \lim_{n\to\infty } \frac{a^{H_n}}{1+n} ,\quad a>0 $
A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1?
[i]Proposed by Andy Xu[/i]
$M,N$ are midpoints of $AB$ and $CD$ for convex quadrilateral $ABCD$. Points $X$ and $Y$ are on $ AD$ and $BC$ and $XD=3AX,YC=3BY$. $\angle MXA=\angle MYB = 90$.
Prove that $\angle XMN=\angle ABC$
Let $m,n\in\mathbb Z_{\ge 0},$ $a_0,a_1,\ldots ,a_m,b_0,b_1,\ldots ,b_n\in\mathbb R_{\ge 0}$ For any integer $0\le k\le m+n,$ define $c_k:=\max_{i+j=k}a_ib_j.$ Proof
$$\frac 1{m+n+1}\sum_{k=0}^{m+n}c_k\ge\frac 1{(m+1)(n+1)}\sum_{i=0}^{m}a_i\sum_{j=0}^{n}b_j.$$
[i]Created by Yinghua Ai[/i]
In a square $ABCD$, $E$ is the midpoint of side $BC$. Line $AE$ intersects line $DC$ at $F$ and diagonal $BD$ at $G$. If the area $(EFC) = 8$, determine the area $(GBE)$.
Let $\left( A,+, \cdot \right)$ be a ring that verifies the following properties:
(i) it has a unit, $1$, and its order is $p$, a prime number;
(ii) there is $B \subset A, \, |B| = p$, such that: for all $x,y \in A$, there is $b \in B$ such that $xy = byx$.
Prove that $A$ is commutative.
[i]Ion Savu[/i]
Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$
Alice and Bob play a game. First, Alice secretly picks a finite set $S$ of lattice points in the Cartesian plane. Then, for every line $\ell$ in the plane which is horizontal, vertical, or has slope $+1$ or $-1$, she tells Bob the number of points of $S$ that lie on $\ell$. Bob wins if he can determine the set $S$.
Prove that if Alice picks $S$ to be of the form
\[S = \{(x, y) \in \mathbb{Z}^2 \mid m \le x^2 + y^2 \le n\}\]
for some positive integers $m$ and $n$, then Bob can win. (Bob does not know in advance that $S$ is of this form.)
[i]Proposed by Mark Sellke[/i]
Let $\triangle ABC$ be an acute triangle. Point $D$ is arbitrarily chosen on the side $BC$. Let the circumcircle of the triangle $\triangle ADB$ intersect the segment $AC$ at $M$, and the circumcircle of the triangle $\triangle ADC$ intersect the segment $AB$ at $N$. Prove that the tangents of the circumcircle of the triangle $\triangle AMN$ at $M$ and $N$ intersect at a point that lies on the line $BC$.
In $R^d$ , all $n^d$ points of an n × n × ··· × n cube grid are contained in 2n - 3 hyperplanes. Prove that n ($n\geq3$) hyperplanes can be chosen from these so that they contain all points of the grid.
Let $H$ be the orthocenter of triangle $ABC$, and $AD$, $BE$, $CF$ be the three altitudes of triangle $ABC$. Let $G$ be the orthogonal projection of $D$ onto $EF$, and $DD'$ be the diameter of the circumcircle of triangle $DEF$. Line $AG$ and the circumcircle of triangle $ABC$ intersect again at point $X$. Let $Y$ be the intersection of $GD'$ and $BC$, while $Z$ be the intersection of $AD'$ and $GH$. Prove that $X$, $Y$, and $Z$ are collinear.
[i]Proposed by Li4 and Untro368.[/i]
In isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\vartriangle ACM$ with the circle with center $B$ passing through $M$, such that $P \ne M$. If $\angle BPC = 135^o$, then $\frac{CP}{AP}$ can be written as $a +\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime. Find $a + b$.
Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.
In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$.
Prove that $BD + DA = BC$.
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.
Given is a segment $AB$. Three points $X, Y, Z$ are picked in the space so that $ABX$ is an equilateral triangle and $ABYZ$ is a square. Prove that the orthocenters of all triangles $XYZ$ obtained in this way belong to a fixed circle.
[i]Alexandr Matveev[/i]
Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.
Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.
Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that:
[list]
[*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and
[*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$.
[/list]
[i]Allen Wang[/i]
Suppose July of year $ N$ has five Mondays. Which of the following must occur five times in August of year $ N$? (Note: Both months have $ 31$ days.)
$ \textbf{(A)}\ \text{Monday} \qquad
\textbf{(B)}\ \text{Tuesday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Thursday} \qquad
\textbf{(E)}\ \text{Friday}$
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.
[i]Proposed by Paisan Nakmahachalasint, Thailand[/i]
Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that:
a) $(AB)^2 = AB.$
b) $(AB - BA)^3 = O_n.$