We have $3$ circles such that any $2$ of them are externally tangent. Let $a$ be length of the outer tangent common to a pair of them. The lengths $b$ and $c$ are defined similarly. If $T$ is the sum of the areas of such circles, show that $\pi (a + b + c)^2 \le 12T $. Note: In In the case of externally tangent circles, the common external tangent is the segment tangent to them that touches them at different points.

Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}$, where $ [x]$ denotes the floor function. For every $ k \ge 0$, find the number $ n(k)$ of positive integers $ n$ such that $ 2^k \le n < 2^{k+1}$ and $ a_n=0$.

Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table or it is outside the table (two $Z$ shapes can overlap and $Z$ shapes can rotate)?

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy \[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\] where $a,b$, and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$.

Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$$

Which of the following acceleration [i]vs.[/i] time graphs most closely represents the acceleration of the toy car? [asy] size(300); Label f; f.p=fontsize(8); xaxis(0,3); yaxis(-2,2); label("Time (s)",(2.8,0),0.5*N); label(rotate(90)*"Acceleration",(-0.2,0),W); label("$0$",(-0,0),SW,fontsize(9)); label("1",(1,0),2*S,fontsize(9)); label("2",(2,0),2*S,fontsize(9)); label("3",(3,0),2*S,fontsize(9)); draw((0.5,0)--(0.5,-0.1)); draw((1,0)--(1,-0.1)); draw((1.5,0)--(1.5,-0.1)); draw((2,0)--(2,-0.1)); draw((2.5,0)--(2.5,-0.1)); draw((3,0)--(3,-0.1)); label("(a)",(1.5,-2),N); pair A, B, C, D, E, F; A = (0,1); B = (1,1); C = (1,0); D = (1.5,0); E = (1.5, 0.5); F = (3, 0.5); draw(A--B--C--D--E--F); real x=6; Label f; f.p=fontsize(8); draw((x+3,0)--(x+0,0)); draw((x,-2)--(x,2)); label("Time (s)",(x+2.8,0.03),0.5*N); label(rotate(90)*"Acceleration",(x-0.2,0),W); label("$0$",(x+0,0),SW,fontsize(9)); label("1",(x+1,0),2*S,fontsize(9)); label("2",(x+2,0),2*S,fontsize(9)); label("3",(x+3,0),2*S,fontsize(9)); draw((x+0.5,0)--(x+0.5,-0.1)); draw((x+1,0)--(x+1,-0.1)); draw((x+1.5,0)--(x+1.5,-0.1)); draw((x+2,0)--(x+2,-0.1)); draw((x+2.5,0)--(x+2.5,-0.1)); draw((x+3,0)--(x+3,-0.1)); label("(b)",(x+1.5,-2),N); /*The lines*/ pair G, H, I, J, K, L; G = (x+0,1); H = (x+1,1); I = (x+1,0); J = (x+1.5,0); K = (x+1.5, -0.5); L = (x+3, -0.5); draw(G--H--I--J--K--L);[/asy][asy] size(300); Label f; f.p=fontsize(8); xaxis(0,3); yaxis(-2,2); label("Time (s)",(2.8,0),0.5*N); label(rotate(90)*"Acceleration",(-0.1,0),W); label("$0$",(-0,0),SW,fontsize(9)); label("1",(1,0),2*S,fontsize(9)); label("2",(2,0),2*S,fontsize(9)); label("3",(3,0),2*S,fontsize(9)); draw((0.5,0)--(0.5,-0.1)); draw((1,0)--(1,-0.1)); draw((1.5,0)--(1.5,-0.1)); draw((2,0)--(2,-0.1)); draw((2.5,0)--(2.5,-0.1)); draw((3,0)--(3,-0.1)); label("(c)",(1.5,-2),N); pair A, B, C, D, E, F; A = (0,0.5); B = (1,0.5); C = (1,0); D = (1.5,0); E = (1.5, -1); F = (3, -1); draw(A--B--C--D--E--F); real x = 6; Label f; f.p=fontsize(8); draw((x+3,0)--(x+0,0)); draw((x,-2)--(x,2)); label("Time (s)",(x+3.4,0),0.5*N); label(rotate(90)*"Acceleration",(x-0.2,0),W); label("$0$",(x+0,0),SW,fontsize(9)); label("1",(x+1,0),2*S,fontsize(9)); label("2",(x+2,0),2*S,fontsize(9)); label("3",(x+3,0),2*S,fontsize(9)); draw((x+0.5,0)--(x+0.5,-0.1)); draw((x+1,0)--(x+1,-0.1)); draw((x+1.5,0)--(x+1.5,-0.1)); draw((x+2,0)--(x+2,-0.1)); draw((x+2.5,0)--(x+2.5,-0.1)); draw((x+3,0)--(x+3,-0.1)); label("(d)",(x+1.5,-2),N); /*The lines*/ pair K, L, M, N, O, P, Q, R; K = (x+0,1); L = (x+1,1); M = (x+1,0.5); N= (x+1.5,0.5); O= (x+1.5, -0.5); P = (x+2.5, -0.5); Q = (x+2.5, 0.5); R = (x+3, 0.5); draw(K--L--M--N--O--P--Q--R);[/asy][asy] size(150); Label f; f.p=fontsize(8); xaxis(0,3); yaxis(-2,2); label("Time (s)",(3.2,0.03),N); label(rotate(90)*"Acceleration",(-0.1,0),W); label("$0$",(-0,0),SW,fontsize(9)); label("1",(1,0),2*S,fontsize(9)); label("2",(2,0),2*S,fontsize(9)); label("3",(3,0),2*S,fontsize(9)); draw((0.5,0)--(0.5,-0.1)); draw((1,0)--(1,-0.1)); draw((1.5,0)--(1.5,-0.1)); draw((2,0)--(2,-0.1)); draw((2.5,0)--(2.5,-0.1)); draw((3,0)--(3,-0.1)); label(rotate(90)*"Acceleration",(-0.1,0),W); label("(e)",(1.5,-2),N); /*The lines*/ pair A, B, C, D, E, F, G, H; A = (0,1); B = (1,1); C = (1,0.5); D = (1.5,0.5); E = (1.5, -0.5); F = (2.5, -0.5); G = (2.5, 0.5); H = (3, 0.5); draw(A--B--C--D--E--F--G--H); [/asy]

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Amy and Bob choose numbers from $0,1,2,\cdots,81$ in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all $82$ numbers are chosen, let $A$ be the sum of all the numbers Amy chooses, and let $B$ be the sum of all the numbers Bob chooses. During the process, Amy tries to make $\gcd(A,B)$ as great as possible, and Bob tries to make $\gcd(A,B)$ as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine $\gcd(A,B)$ when all $82$ numbers are chosen.

In triangle $ABC$, the angle bisectors $AD$, $BE$ and $CF$ are drawn, intersecting at point $I$. The perpendicular bisector of the segment $AD$ intersects lines $BE$ and $CF$ at points $M$ and $N$, respectively. Prove that points $A$, $I$, $M$ and $ N$ lie on the same circle.

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

Let be given a convex polyhedron whose all faces are triangular. The vertices of the polyhedron are colored using three colors. Prove that the number of faces with vertices in all the three colors is even.

We are given the equation $x^3-cx^2+(c-3)x+1=0$, where $c$ is an arbitrary number. Prove that, if the equation has at least one rational root, then all of its roots are rational.

There is a circle $k$ in a plane with center $M$ and radius $r$. The following illustration, through which every point $P \ne M$., is called a “reflection on the circle $k$” from $\varepsilon$ a point $P'$ from $\varepsilon$ is assigned: (1) $P'$ lies on the ray emanating from$ M$ and passing through $P$. (2) It is $MP \cdot MP' = r^2$. a) Construct the mirror point $ P'$ for any given point $P \ne M$ inside $k$. b) Let another circle $k_1$ be given arbitrarily, but such that $M$ lies outside $k_1$.Construct $k'_1$ , i.e. the set of all mirror points $P'$ of the points $P$ of $k_1$.

Find the least value of $k\in \mathbb{N}$ with the following property: There doesn’t exist an arithmetic progression with 2019 members, from which exactly $k$ are integers.

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

In convex quadrilateral $AEBC$, $\angle BEA = \angle CAE = 90^{\circ}$ and $AB = 15$, $BC = 14$ and $CA = 13$. Let $D$ be the foot of the altitude from $C$ to $\overline{AB}$. If ray $CD$ meets $\overline{AE}$ at $F$, compute $AE \cdot AF$. [i]Proposed by David Stoner[/i]

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Define binary operations $\diamondsuit$ and $\heartsuit$ by $$a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}$$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsuit\, 2$ and $$a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}$$ for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$? $\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$

If $ \theta$ is an acute angle and $ \sin \frac12 \theta \equal{} \sqrt{\frac{x\minus{}1}{2x}}$, then $ \tan \theta$ equals $ \textbf{(A)}\ x \qquad \textbf{(B)}\ \frac1{x} \qquad \textbf{(C)}\ \frac{\sqrt{x\minus{}1}}{x\plus{}1} \qquad \textbf{(D)}\ \frac{\sqrt{x^2\minus{}1}}{x} \qquad \textbf{(E)}\ \sqrt{x^2\minus{}1}$

Let be a convex quadrilateral $ ABCD $ and the points $ M,N,P,Q $ such that $ MAB\sim NBC\sim PCD\sim QDA. $ [b]a)[/b] Prove that $ ABCD $ is a parallelogram if and only if $ MNPQ $ is a parallelogram. [b]b)[/b] Show that if the diagonals of $ MNPQ $ are congruent and perpendicular, then the diagonals of $ ABCD $ are congruent and perpendicular, or $ MAB $ is a right isosceles triangle. [i]Nelu Chichirim[/i]

Milly chooses a positive integer $n$ and then Uriel colors each integer between $1$ and $n$ inclusive red or blue. Then Milly chooses four numbers $a, b, c, d$ of the same color (there may be repeated numbers). If $a+b+c= d$ then Milly wins. Determine the smallest $n$ Milly can choose to ensure victory, no matter how Uriel colors.

Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$. Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$. (a) Prove that $L$ is nonempty. (b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$. [i]Proposed by David Yang[/i]