Let $ABC$ be an acute triangle and $G$ be the intersection of the meadians of triangle $ABC$. Let $D $be the foot of the altitude drawn from $A$ to $BC$. Draw a parallel line such that it is parallel to $BC$ and one of the points of it is $A$.Donate the point $S$ as the intersection of the parallel line and circumcircle $ABC$. Prove that $S,G,D$ are co-linear [asy] size(6cm); defaultpen(fontsize(10pt)); pair A = dir(50), S = dir(130), B = dir(200), C = dir(-20), G = (A+B+C)/3, D = foot(A, B, C); draw(A--B--C--cycle, black+linewidth(1)); draw(A--S^^A--D, magenta); draw(S--D, red+dashed); draw(circumcircle(A, B, C), heavymagenta); string[] names = {"$A$", "$B$", "$C$","$D$", "$G$","$S$"}; pair[] points = {A, B, C,D,G,S}; pair[] ll = {A, B, C,D, G,S}; int pt = names.length; for (int i=0; i<pt; ++i) dot(names[i], points[i], dir(ll[i])); [/asy]

A sequence $(a_k)_{k=1}^{\infty}$ has the property that there is a natural number $n$ such that $a_1 + a_2 +...+ a_n = 0$ and $a_{n+k} = a_k$ for all $k$. Prove that there exists a natural number $N$ such that $$\sum_{i=N}^{N+k} a_i \ge 0 \,\, \,\, for \,\,\,\, k = 0,1,2...$$

A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row? 1 2 3 4 5 3 5 7 9 8 12 16 20 28 4

In the diagram below, fill the $12$ circles with numbers from the following bank so that each number is used once. Two circles connected by a single line must contain relatively prime numbers. Two circles connected by a double line must contain numbers that are not relatively prime. $$\text{Bank: } 20, 21, 22, 23, 24, 25, 27, 28, 30 ,32, 33 ,35$$ [asy] real HRT3 = sqrt(3) / 2; void drawCircle(real x, real y, real r) { path p = circle((x,y), r); draw(p); fill(p, white); } void drawCell(int gx, int gy) { real x = 0.5 * gx; real y = HRT3 * gy; drawCircle(x, y, 0.35); } void drawEdge(int gx1, int gy1, int gx2, int gy2, bool doubled) { real x1 = 0.5 * gx1; real y1 = HRT3 * gy1; real x2 = 0.5 * gx2; real y2 = HRT3 * gy2; if (doubled) { real dx = x2 - x1; real dy = y2 - y1; real ox = -0.035 * dy / sqrt(dx * dx + dy * dy); real oy = 0.035 * dx / sqrt(dx * dx + dy * dy); draw((x1+ox,y1+oy)--(x2+ox,y2+oy)); draw((x1-ox,y1-oy)--(x2-ox,y2-oy)); } else { draw((x1,y1)--(x2,y2)); } } drawEdge(2, 0, 4, 0, true); drawEdge(2, 0, 1, 1, true); drawEdge(2, 0, 3, 1, true); drawEdge(4, 0, 3, 1, false); drawEdge(4, 0, 5, 1, false); drawEdge(1, 1, 0, 2, false); drawEdge(1, 1, 2, 2, false); drawEdge(1, 1, 3, 1, false); drawEdge(3, 1, 2, 2, true); drawEdge(3, 1, 4, 2, true); drawEdge(3, 1, 5, 1, false); drawEdge(5, 1, 4, 2, true); drawEdge(5, 1, 6, 2, false); drawEdge(0, 2, 1, 3, false); drawEdge(0, 2, 2, 2, false); drawEdge(2, 2, 1, 3, false); drawEdge(2, 2, 3, 3, true); drawEdge(2, 2, 4, 2, false); drawEdge(4, 2, 3, 3, false); drawEdge(4, 2, 5, 3, false); drawEdge(4, 2, 6, 2, false); drawEdge(6, 2, 5, 3, true); drawEdge(1, 3, 3, 3, true); drawEdge(3, 3, 5, 3, false); drawCell(2, 0); drawCell(4, 0); drawCell(1, 1); drawCell(3, 1); drawCell(5, 1); drawCell(0, 2); drawCell(2, 2); drawCell(4, 2); drawCell(6, 2); drawCell(1, 3); drawCell(3, 3); drawCell(5, 3); [/asy]

How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

Let $ABCD$ be a convex quadrilateral. Construct $S$ and $T$ on the side $AD$ and $AB$ respectively such that $AS=AT$. Construct $U$ and $V$ on the side $BC$ and $CD$ respectively such that $CU=CV$. Assume that $BT=BU$ and $ST, UV, BD$ are concurrent, prove that $AB+CD=BC+AD$.

Show that each number is of the form $$2^{5^{2^{5^{...}}}}+ 4^{5^{4^{5^{...}}}}$$ is divisible by $2008$, where the exponential towers can be any independent ones have height $\ge 3$.

For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$

Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).$$

Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number.

In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.

Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$.

In $\triangle ABC$, $b\neq1$. If $\frac{C}{A}$ and $\frac{\sin B}{\sin A}$ are solutions to equation $\log_{\sqrt{b}}x=\log_{b}(4x-4)$, then $\triangle ABC$ $\text{(A)}$is an isosceles triangle, but not right-angled triangle $\text{(B)}$is a right-angled triangle, but not isosceles triangle $\text{(C)}$is an isosceles right-angled triangle $\text{(D)}$is neither a right-angled triangle nor an isosceles triangle

Let $ABC$ be a triangle inscribed in circle $(O)$, with its altitudes $BH_b, CH_c$ intersect at orthocenter $H$ ($H_b \in AC$, $H_c \in AB$). $H_bH_c$ meets $BC$ at $P$. Let $N$ be the midpoint of $AH, L$ be the orthogonal projection of $O$ on the symmedian with respect to angle $A$ of triangle $ABC$. Prove that $\angle NLP = 90^o$.

Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are $1+2i,-2+i$ and $-1-2i$. The fourth number is A. $2+i$ B. $2-i$ C. $1-2i$ D. $-1+2i$ E. $-2-i$

An equilateral triangle with side 101 is placed on a plane so that one of its sides is horizontal and the triangle is above it. It is divided into smaller equilateral triangles with side 1 by segments parallel to its sides. All sides of these smaller triangles are colored red (including the entire border of the large triangle). An equilateral triangle on a plane is called a "mirror" triangle if its sides are parallel to the sides of the original triangle, but it lies below its horizontal side. What is the smallest number of contours of mirror triangles needed to cover all the red segments? (Mirror triangles may overlap and extend beyond the original triangle.) [i]Proposed by Dmitry Shiryayev[/i]

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$. Is it possible to assert that it is more than $3.5$?

How many four-digit positive integers have at least one digit that is a 2 or a 3? $ \textbf{(A) } 2439 \qquad \textbf{(B) } 4096 \qquad \textbf{(C) } 4903 \qquad \textbf{(D) } 4904 \qquad \textbf{(E) } 5416$

Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$-plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals: $\textbf{(A) }-\tfrac35\qquad \textbf{(B) }-\tfrac25\qquad \textbf{(C) }-\tfrac15\qquad \textbf{(D) }\tfrac15\qquad \textbf{(E) }\text{either }-\tfrac15\text{ or }\tfrac15$

Let $n\ge5$ be an integer. Find the biggest integer $k$ such that there always exists a $n$-gon with exactly $k$ interior right angles. (Find $k$ in terms of $n$).

What is the value of $\dfrac{11!-10!}{9!}$? $\textbf{(A)}\ 99\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 121\qquad\textbf{(E)}\ 132$