Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle BCD = \theta$ for some acute angle $\theta$. Point $X$ lies inside the quadrilateral such that $\angle XAD = \angle XDA = 90^{\circ}-\theta$. Prove that $BX = XC$.

$\overline{AB}$ is a diameter of circle $C_1$. Point $P$ is on $C_1$ such that $AP>BP$. Circle $C_2$ is centered at $P$ with radius $PB$. The extension of $\overline{AP}$ past $P$ meets $C_2$ at $Q$. Circle $C_3$ is centered at $A$ and is externally tangent to $C_2$. Circle $C_4$ passes through $A$, $Q$, and $R$. Find, with proof, the ratio between the area of $C_4$ and the area of $C_1$, and show that this ratio is the same for all points $P$ on $C_1$ such that $AP>BP$.

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

Two points on a circle are joined by a broken line shorter than the diameter of the circle. Prove that there exists a diameter which does not intersect this broken line. (Folklor )

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

Show that [list=a] [*] if $n>49$, then there are positive integers $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}<1$. [*] if $n>4$, then there are $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}>1$.[/list]

On the sides of a convex, non-regular $m$-gon are built externally regular heptagons. It is known that their centers are vertices of a regular $m$-gon. What’s the least possible value of $m$?

For arbitrary natural number $n$, prove that $\sum^n_{k=0}C^k_n2^kC^{[(n-k)/2]}_{n-k}=C^n_{2n+1}$, where $C^0_0=1$ and $[\dfrac{n-k}{2}]$ denotes the integer part of $\dfrac{n-k}{2}$.

Three bells begin to ring simultaneously. The intervals between strikes for these bells are, respectively, $\frac43$ seconds, $\frac53$ second and $2$ seconds. Impacts that coincide in time are perceived as one. How many beats will be heard in $1$ minute? (Include first and last.)

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

$f_{1},f_{2},\dots,f_{n}$ are polynomials with integer coefficients. Prove there exist a reducible $g(x)$ with integer coefficients that $f_{1}+g,f_{2}+g,\dots,f_{n}+g$ are irreducible.

[url=https://artofproblemsolving.com/community/c675547][b]Greece JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663730p10567608][b]Problem 1[/b][/url]. Positive real numbers $a,b,c$ satisfy $a+b+c=1$. Prove that $$(a+1)\sqrt{2a(1-a)} + (b+1)\sqrt{2b(1-b)} + (c+1)\sqrt{2c(1-c)} \geq 8(ab+bc+ca).$$ Also, find the values of $a,b,c$ for which the equality happens. [url=http://artofproblemsolving.com/community/c6h1663731p10567619][b]Problem 2[/b][/url]. Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$. The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$. Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$. [url=http://artofproblemsolving.com/community/c6h1663732p10567627][b]Problem 3[/b][/url]. Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$. [url=http://artofproblemsolving.com/community/c6h1663734p10567640][b]Problem 4[/b][/url]. Let $ABC$ be an equilateral triangle of side length $a$, and consider $D$, $E$ and $F$ the midpoints of the sides $(AB), (BC)$, and $(CA)$, respectively. Let $H$ be the the symmetrical of $D$ with respect to the line $BC$. Color the points $A, B, C, D, E, F, H$ with one of the two colors, red and blue. [list=1] [*] How many equilateral triangles with all the vertices in the set $\{A, B, C, D, E, F, H\}$ are there? [*] Prove that if points $B$ and $E$ are painted with the same color, then for any coloring of the remaining points there is an equilateral triangle with vertices in the set $\{A, B, C, D, E, F, H\}$ and having the same color. [*] Does the conclusion of the second part remain valid if $B$ is blue and $E$ is red? [/list]

$8$ people participate in a party. (1) Among any $5$ people there are $3$ who pairwise know each other. Prove that there are $4$ people who paiwise know each other. (2) If Among any $6$ people there are $3$ who pairwise know each other, then can we find $4$ people who pairwise know each other?

Let $a$ and $b$ be positive integers with $a>b$. Suppose that $$\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}}$$ is an integer. (a) Must $\sqrt{a}$ be an integer? (b) Must $\sqrt{b}$ be an integer?

Let $A, B, C,$ be points in the plane, such that we can draw $3$ equal circumferences in which the first one passes through $A$ and $B$, the second one passes through $B$ and $C$, the last one passes through $C$ and $A$, and all $3$ circumferences share a common point $P$. Show that the radius of each of these circumferences is equal to the circumradius of triangle $ABC$, and that $P$ is the orthocenter of triangle $ABC$.

The War of $1812$ started with a declaration of war on Thursday, June $18$, $1812$. The peace treaty to end the war was signed $919$ days later, on December $24$, $1814$. On what day of the week was the treaty signed? $\textbf{(A)}\ \text{Friday} \qquad \textbf{(B)}\ \text{Saturday} \qquad \textbf{(C)}\ \text{Sunday} \qquad \textbf{(D)}\ \text{Monday} \qquad \textbf{(E)}\ \text{Tuesday} $

The number $2.5252525...$ can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is: $ \textbf{(A) }7\qquad\textbf{(B)} 29\qquad\textbf{(C) }141\qquad\textbf{(D) }349\qquad\textbf{(E) }\text{none of these} $

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(xy) + xy$ for all $x, y \in R$.

Let $S$ be the subset of $N$($N$ is the set of all natural numbers) satisfying: i)Among each $2003$ consecutive natural numbers there exist at least one contained in $S$; ii)If $n \in S$ and $n>1$ then $[\frac{n}{2}] \in S$ Prove that:$S=N$ I hope it hasn't posted before. :lol: :lol:

Denote by $ d(n)$ the number of all positive divisors of a natural number $ n$ (including $ 1$ and $ n$). Prove that there are infinitely many $ n$, such that $ n/d(n)$ is an integer.

The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let $$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and $$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that $$|D|\le2|S|+\frac{n^2}2.$$

In how many ways can $n^2$ distinct real numbers be arranged into an $n\times n$ array $(a_{ij })$ such that max$_{j}$ min $_i \,\, a_{ij} $= min$_i$ max$_j \,\, a_{ij}$?

Given a triangle with sides $a, b, c$ that satisfy $\frac{a}{b+c}=\frac{c}{a+b}$. Determine the angles of this triangle, if you know that one of them is equal to $120^0$.

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.