Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \[a_1 + k, a_2 + 2k, \dots, a_p + pk\] produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$. [i]Proposed by Ankan Bhattacharya[/i]

Given a triangle $ABC$ with $\angle C = 90^o$. a) Construct the circle with center $C$, so that one of the tangents from $A$ to that circle is parallel to one of the tangents from $B$ to that circle. b) A circle with center $C$ has two parallel tangents passing through A and go respectively. If $AC = b$ and $BC = a$, express the radius of the circle in terms of $a$ and $b$.

On the sides $PQ, QR, RP$ of $\vartriangle PQR$ segments $AB, CD, EF$ are drawn. Given a point $S_0$ inside triangle $\vartriangle PQR$, find the locus of points $S$ for which the sum of the areas of triangles $\vartriangle SAB$, $\vartriangle SCD$ and $\vartriangle SEF$ is equal to the sum of the areas of triangles $\vartriangle S_0AB$, $\vartriangle S_0CD$, $\vartriangle S0EF$. Consider separately the case $$\frac{AB}{PQ }= \frac{CD}{QR} = \frac{EF}{RP}.$$

A non-equilateral triangle $ABC$ is inscribed in a circle $\Gamma$ with centre $O$, radius $R$ and its incircle has centre $I$ and touches $BC,CA,AB$ at $D,E, F$, respectively. A circle with centre $I$ and radius $r$ intersects the rays $[ID), [IE), [IF)$ at $A',B',C'$. Show that the orthocentre $K$ of $\vartriangle A'B'C'$ is on the line $OI$ and that $\frac{IK}{IO}=\frac{r}{R}$ Michel Bataille .

Consider $5$ distinct positive integers. Can their mean be a)Exactly $3$ times larger than their largest common divisor? b)Exactly $2$ times larger than their largest common divisor?

Find the number of ordered pairs of integers $(a, b)$ such that the sequence $$3, 4, 5, a, b, 30, 40, 50$$ is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Suppose that [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$a$",(1,sqrt(3)-0.2),S); label("$b$",(sqrt(3)/10,0.1),ENE); label("$c$",(2-sqrt(3)/10,0.1),WNW); [/asy] means $a+b-c$. For example, [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$5$",(1,sqrt(3)-0.2),S); label("$4$",(sqrt(3)/10,0.1),ENE); label("$6$",(2-sqrt(3)/10,0.1),WNW); [/asy] is $5+4-6 = 3$. Then the sum [asy] unitsize(18); draw((0,0)--(2,0)--(1,sqrt(3))--cycle); label("$1$",(1,sqrt(3)-0.2),S); label("$3$",(sqrt(3)/10,0.1),ENE); label("$4$",(2-sqrt(3)/10,0.1),WNW); draw((3,0)--(5,0)--(4,sqrt(3))--cycle); label("$2$",(4,sqrt(3)-0.2),S); label("$5$",(3+sqrt(3)/10,0.1),ENE); label("$6$",(5-sqrt(3)/10,0.1),WNW); label("$+$",(2.5,-0.1),N); [/asy] is $\text{(A)}\ -2 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$

$ \left|AC\right| \equal{} 8 \sqrt {2}$, $ B$ is the midpoint of $ \left[AC\right]$, $ E$ is the midpoint of arc $ AB$ of a circle having chord $ \left[AB\right]$, and $ D$ is the point of tangency drawing from $ C$.($ D$ lies on the opposite side of line $ AB$ to $ E$). If $ \left[DE\right]\bigcap \left[AB\right] \equal{} \left\{F\right\}$, $ \left|CF\right| \equal{} ?$ $\textbf{(A)}\ 5\sqrt {2} \qquad\textbf{(B)}\ 4\sqrt {2} \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 4\sqrt {3}$

Find the number of ways to distribute $11$ balls into $5$ boxes with different sizes, so that each box receives at least one ball, and the total number of balls in the largest and smallest boxes is more than the total number of balls in the remaining boxes.

Do there exist real numbers $a$ and $b$ such that [list=a][*] $a+b$ is rational and $a^n +b^n $ is irrational for all $n \in \mathbb{N}$ with $n \ge 2$? [*] $a+b$ is irrational and $a^n +b^n $ is rational for all $n \in \mathbb{N}$ with $n \ge 2$?[/list]

There are $N$ points on the plane; no three of them belong to the same straight line. Every pair of points is connected by a segment. Some of these segments are colored in red and the rest of them in blue. The red segments form a closed broken line without self-intersections(each red segment having only common endpoints with its two neighbors and no other common points with the other segments), and so do the blue segments. Find all possible values of $N$ for which such a disposition of $N$ points and such a choice of red and blue segments are possible.

Let $a, b$ and $c$ be positive real numbers such that $a+b+c = abc$. Prove that at least one of $a, b$ or $c$ is greater than $\frac{17}{10}$ .

Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$

For any polynomial f, denote by $P_f$ the number of integers n for which f(n) is a (positive) prime number. Let $q_d = max P_f$ , where f runs over all polynomials with integer coefficients with degree d and reducible over $\mathbb{Q}$. Prove that $\forall d\geq 2$ , $q_d = d$.

Let $ABCD$ be a trapezoid with the longer base $AB$ such that its diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of the triangle $ABC$ and $E$ be the intersection of the lines $OB$ and $CD$. Prove that $BC^2 = CD \cdot CE$.

Jia Herng has a circle $\omega$ with center $O$, and $P$ is a point outside of $\omega$. Let $PX$ and $PY$ are two lines tangent to $\omega$ at $X$ and $Y$ , and $Q$ is a point on segment $PX$. Let $R$ is a point on the ray $PY$ beyond $Y$ such that $QX = RY$. Help Jia Herng prove that the points $O$, $P$, $Q$, $R$ are concyclic.

A rational number between $ \sqrt{2}$ and $ \sqrt{3}$ is: $ \textbf{(A)}\ \frac{\sqrt{2} \plus{} \sqrt{3}}{2} \qquad \textbf{(B)}\ \frac{\sqrt{2} \cdot \sqrt{3}}{2}\qquad \textbf{(C)}\ 1.5\qquad \textbf{(D)}\ 1.8\qquad \textbf{(E)}\ 1.4$

$A,B,C,D,E$ lie on $\odot O$ in that order,and $$BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J.$$ Prove that $\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD}$ when and only when $F,G,H,I,J$ are concyclic.

G. H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: “I came here in taxi-cab number $1729$. That number seems dull to me, which I hope isn’t a bad omen.” “Nonsense,” said Ramanujan. “The number isn’t dull at all. It’s quite interesting. It’s the smallest number that can be expressed as the sum of two cubes in two different ways.” Ramujan had immediately seen that $1729=12^3+1^3=10^3+9^3$. What is the smallest positive integer representable as the sum of the cubes of [i]three[/i] positive integers in two different ways?

Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.

Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy $$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$

Let $K$ be a point inside an acute triangle $ ABC $, such that $ BC $ is a common tangent of the circumcircles of $ AKB $ and $ AKC$. Let $ D $ be the intersection of the lines $ CK $ and $ AB $, and let $ E $ be the intersection of the lines $ BK $ and $ AC $ . Let $ F $ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $ F$ and radius $FD$ intersect at points $P$ and $Q$. Prove that the segment $PQ$ is a diameter of $k$.

Find all $(x,y)$ real numbers pairs such that, $$x^7+y^7=x^4+y^4$$

Determine alle positive integers $n>1$ with the following property: For each colouring of the lattice points in the plane with $n$ colours, there are three lattice points of the same colour forming an isosceles right triangle with legs parallel to the coordinate axes.