Siva has the following expression, which is missing operations: $$\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than $\frac12$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.

Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions: i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$; ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$. Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.

Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$, $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$, the inequality \[ \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r \] is true for all $r\geq \rho (n)$.

$g$ and $n$ are natural numbers such that $gcd(g^2-g,n)=1$ and $A=\{g^i|i \in \mathbb N\}$ and $B=\{x\equiv (n)|x\in A\}$(by $x\equiv (n)$ we mean a number from the set $\{0,1,...,n-1\}$ which is congruent with $x$ modulo $n$). if for $0\le i\le g-1$ $a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$ prove that $g-1|\sum_{i=0}^{g-1}ia_i$.( the symbol $|$ $|$ means the number of elements of the set)($\frac{100}{6}$ points) the exam time was 4 hours

Let $A$ be the area of the region in the first quadrant bounded by the line $y = \frac{x}{2}$, the x-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx,$ the y-axis, and the ellipse $\frac{x^2}{9} + y^2 = 1.$

Prove that there exists a function $f\colon [\omega_1]^2 \rightarrow \omega _1$ such that (i) $f(\alpha, \beta)< \mathrm{min}(\alpha, \beta)$ whenever $\mathrm{min}(\alpha,\beta)>0$; and (ii) if $\alpha_0<\alpha_1<\ldots<\alpha_i<\ldots<\omega_1$ then $\sup\left\{ a_i \colon i<\omega \right\} =\sup \left\{ f(\alpha_i, \alpha_j)\colon i,j<\omega\right\}$.

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

Let $n\ge 2$ and $d\ge 1$ be integers with $d\mid n$, and let $x_1,x_2,\ldots x_n$ be real numbers such that $x_1+x_2+\cdots + x_n=0$. Show that there are at least $\binom{n-1}{d-1}$ choices of $d$ indices $1\le i_1<i_2<\cdots <i_d\le n $ such that $x_{i_{1}}+x_{i_{2}}+\cdots +x_{i_{d}}\ge 0$.

The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$. Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.

Assume $a,b,c$ are odd integers. Show that the quadratic equation \[ ax^2 + bx + c = 0 \] has no rational solutions. (A number is said to be [i]rational[/i], if it can be written as a fraction: $\frac{\text{integer}}{\text{integer}}$.)

Set $|A|=n$. $A_1,A_2,\cdots,A_m$ are subsets of $A$, and $A_i\not\subseteq A_j$ for any $1\leq i<j\leq m$. Prove: [b](a)[/b] $\sum_{i=1}^{m}\frac{1}{\text{C}_n^{|A_i|}}\leq1$. [b](b)[/b] $\sum_{i=1}^{m}\text{C}_n^{|A_i|}\geq m^2$.

A set $M$ of real numbers will be called [i]special [/i] if it has the properties: (i) for each $x, y \in M, x\ne y$, the numbers $x + y$ and $xy$ are not zero and exactly one of them is rational; (ii) for each $x \in M, x^2$ is irrational. Find the maximum number of elements of a [i]special [/i] set.

Prove that for four positive real numbers $a,b,c,d$ the following inequality holds and find all equality cases: $$a^3 +b^3 +c^3 +d^3 \geq a^2 b +b^2 c+ c^2 d +d^2 a.$$

Let $ABC$ be a triangle, let $H$ be the orthocentre and $L,M,N$ the midpoints of the sides $AB, BC, CA$ respectively. Prove that \[HL^{2} + HM^{2} + HN^{2} < AL^{2} + BM^{2} + CN^{2}\] if and only if $ABC$ is acute-angled.

A polygonal line with a finite number of segments has all its vertices on a parabola. Any two adjacent segments make equal angles with the tangent to the parabola at their point of intersection. One end of the polygonal line is also on the axis of the parabola. Show that the other vertices of the polygonal line are all on the same side of the axis.

A red coin is placed in a cell of $2n \times 2n$ board. Every move it can either move like a bishop and change its color (red to blue, blue to red), or move like a knight and not change its color. After some time the coin has visited every cell exactly twice. Prove that the number of cells in which the coin was both red and blue is even. [i]M. Zorka[/i]

Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .

Prove that for any $\delta$ greater than 1 and any positive number $\epsilon$, there is an $n$ such that $\left \vert \frac{\sigma (n)}{n} -\delta \right \vert < \epsilon$.

Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles. (C.Pohoata)

Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$ are positive real numbers, than $ \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}a_{j}}{c_{i} \plus{} c_{j}}\right)\left(\sum_{i,j \equal{} 1}^{n}\frac {b_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)\ge \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)^{2}$.

In parallelogram $ABCD,$ $E$ is a point on $\overline{AD}$ such that $\overline{CE} \perp \overline{AD},$ $F$ is a point on $\overline{CD}$ such that $\overline{AF} \perp \overline{CD},$ and $G$ is a point on $\overline{BC}$ such that $\overline{AG} \perp \overline{BC}.$ Let $H$ be a point on $\overline{GF}$ such that $\overline{AH} \perp \overline{GF},$ and let $J$ be the intersection of lines $EF$ and $BC.$ Given that $AH=8, AE=6,$ and $EF=4,$ compute $CJ.$

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]

For a convex hexagon $AHSIMC$ whose side lengths are all $1$, let $Z$ and $z$ be the maximum and minimum values, respectively, of the three diagonals $AI$, $HM$, and $SC$. If $\sqrt{x}\le Z \le \sqrt{y} $ and $\sqrt{q}\le z \le \sqrt{r} $ , find the product $qrxy$, if $q$,$ r$, $x$, and $y$ are all integers.

Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how many positions could the banana be?

Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$, define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$ For segment $MN$, define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$. Find the smallest real number $\lambda$, such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$, satisfying: for any two points $M,N$ on $l$, $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$