Let $n$ be a positive integer and $a=2^n\cdot 7^{n+1}+11$ and $b=2^{n+1}\cdot 7^n+3$. $a)$ Prove that fraction $\frac{a}{b}$ is irreducible $b)$ Prove that number $a+b-7$ is not a perfect square for any positive integer $n$

Four grasshoppers sit at the vertices of a square. Every second, one of them jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to the point $X'$, then $X$, $Y$ and $X'$ lie on a straight line and $XY = YX'$). Prove that after several jumps no three grasshoppers can be: (a) on a line parallel to a side of the square, (b) on a straight line. (AK Kovaldzhy)

Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.

Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?

Simplify $\sqrt[4]{17 + 12\sqrt2} - \sqrt[4]{17 - 12\sqrt2}$.

The numbers $2,3,7$ have the property that the product of any two of them increased by $1$ is divisible by the third number. Prove that this triple of integer numbers greater than $1$ is the only triple with the given property.

Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$. If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$. [i]Proposed by Andy Xu[/i]

Let $n\ge 3$ be a positive integer. Danielle draws a math flower on the plane Cartesian as follows: first draw a unit circle centered on the origin, then draw a polygon of $n$ vertices with both rational coordinates on the circumference so that it has two diametrically opposite vertices, on each side draw a circumference that has the diameter of that side, and finally paints the area inside the $n$ small circles but outside the unit circle. If it is known that the painted area is rational, find all possible polygons drawn by Danielle.

Prove that for every two positive integers $a,b$ greater than $1$. there exists infinitly many $n$ such that the equation $\phi(a^n-1)=b^m-b^t$ can't hold for any positive integers $m,t$.

A perfect number, greater than $28$ is divisible by $7$. Prove that it is also divisible by $49$.

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$. [i]Proposed by Lewis Chen [/i]

Show that if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},$ then also \[\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},\] provided $n$ is an odd positive integer.

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 \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]