Let $n \geq 2$ be a positive integer. In a mathematics competition, there are $n+1$ students, with one of them being a hacker. The competition is conducted as follows: each receives the same problem with an open-ended answer, has 5 minutes to give their own answer, after which all answers are submitted simultaneously, the correct answer is announced, then they receive a new problem, and so on. The hacker cheats by using spy cameras to see the answers of the other participants. A correct answer gives 1 point, while a wrong answer gives -1 point to everyone except the hacker; for him, it's 0 points because he managed to hack the scoring system. Prove that regardless of the total number of problems, if at some point the hacker is ahead of the second-place contestant by at least $2^{n-2} + 1$ points, then he has a strategy to ensure he will be the sole winner by the end of the competition.

Let $ABC$ be an isosceles triangle $(AB=AC)$ so that $\angle A< 2 \angle B$ . Let $D,Z $ points on the extension of height $AM$ so that $\angle CBD = \angle A$ and $\angle ZBA = 90^\circ$. Let $E$ the orthogonal projection of $M$ on height $BF$, and let $K$ the orthogonal projection of $Z$ on $AE$. Prove that $ \angle KDZ = \angle KDB = \angle KZB$.

You wish to color every vertex, edge, face, and the interior of a cube one color each such that no two adjacent objects are the same color. Faces are adjacent if they share an edge. Edges are adjacent if they share a vertex. The interior is adjacent to all of its faces, edges, and vertices. Each face is adjacent to all of its edges and vertices, but is not adjacent to any other edges or vertices. Each edge is adjacent to both of its vertices, but is not adjacent to any other vertices. What is the minimum number of colors required for a coloring satisfying this property?

For all real numbers $r$, denote by $\{r\}$ the fractional part of $r$, i.e. the unique real number $s\in[0,1)$ such that $r-s$ is an integer. How many real numbers $x\in[1,2)$ satisfy the equation $\left\{x^{2018}\right\} = \left\{x^{2017}\right\}?$

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Let $\lambda \geq 1$ be a real number and $n$ be a positive integer with the property that $\lfloor \lambda^{n+1}\rfloor, \lfloor \lambda^{n+2}\rfloor ,\cdots, \lfloor \lambda^{4n}\rfloor$ are all perfect squares$.$ Prove that $\lfloor \lambda \rfloor$ is a perfect square$.$

Let $S$ be a subset of the integers $1,2,\ldots,100$ that has the property that none of its members is $3$ times another. What is the largest number of members $S$ can have? $\text{(A) }67\qquad\text{(B) }71\qquad\text{(C) }72\qquad\text{(D) }76\qquad\text{(E) }77$

Given are $N$ points on the plane such that no three of them are collinear,which are coloured red,green and black.We consider all the segments between these points and give to each segment a [i]"value"[/i] according to the following conditions: [b]i.[/b]If at least one of the endpoints of a segment is black then the segment's [i]"value"[/i] is $0$. [b]ii.[/b]If the endpoints of the segment have the same colour,re or green,then the segment's [i]"value"[/i] is $1$. [b]iii.[/b]If the endpoints of the segment have different colours but none of them is black,then the segment's [i]"value"[/i] is $-1$. Determine the minimum possible sum of the [i]"values"[/i] of the segments.

$a)$ Prove that there exists $5$ nonnegative real numbers with sum equal to $1$, such that no matter how we arrange them on circle, two neighboring numbers exist with product not less than $\frac{1}{9}$ $a)$ Prove that for every $5$ nonnegative real numbers with sum equal to $1$, we can arrange them on circle, such that product of every two neighboring numbers is not greater than $\frac{1}{9}$

Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation \[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \] has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.

The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Find all quadruples of real numbers $(a,b,c,d)$ satisfying the system of equations \[\begin{cases}(b+c+d)^{2010}=3a\\ (a+c+d)^{2010}=3b\\ (a+b+d)^{2010}=3c\\ (a+b+c)^{2010}=3d\end{cases}\]

Let $n>1$ be a positive integer. Show that the number of residues modulo $n^2$ of the elements of the set $\{ x^n + y^n : x,y \in \mathbb{N} \}$ is at most $\frac{n(n+1)}{2}$. [I]Proposed by N. Safaei (Iran)[/i]

We partition the $n$-element subsets of an $n^2+n-1$-element set into two classes. Prove that one of the classes contains $n$-many pairwise disjunct sets. (translated by Miklós Maróti)

For what natural numbers $n$ for any numbers $a, b , c$, which are values of the angles of an acute triangle, the following inequality is true: $$\sin na + \sin nb + \sin nc < 0?$$

Let $ABCD$ be a convex quadrilateral and let $M$, $N$, $P$ and $Q$ be the midpoints of the sides $AB$, $CD$, $BC$ and $DA$ respectively. The line $MN$ intersects the segments $AP$ and $CQ$ at points $X$ and $Y$, respectively. Suppose that $MX = NY$. Prove that $\text{area}(ABCD) = 4 \cdot \text{area}(BXDY).$

Find the possible coordinates of the vertices of a triangle of which we know that the coordinates of its orthocenter are $ (-3,10), $ those of its circumcenter is $ (-2,-3), $ and those of the midpoint of some side is $ (1,3). $

We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$. Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge? [img]https://i.imgur.com/AHqHHP6.png[/img]

Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ have radii of $20$ and $16$, respectively, find $\sqrt{1+BC/BD}$.

Answer the following questions : $\textbf{(a)}~$ A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2,\cdots, a_k$, each $a_i>1$, such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1$$ Show that if $k$ is stable, then $(k+1)$ is also stable. Using this or otherwise, find all stable numbers. $\textbf{(b)}$ Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^*(y):=\max_{x\in A} \left\{yx-f(x)\right\}$$ whenever the above maximum is finite. For the function $f(x)=\ln x$, determine the set of points for which $f^*$ is defined and find an expression for $f^*(y)$ involving only $y$ and constants.

Which of the following is satisfied by all numbers $ x$ of the form \[ x \equal{} \frac {a_1}{3} \plus{} \frac {a_2}{3^2} \plus{} \cdots \plus{} \frac {a_{25}}{3^{25}},\] where $ a_1$ is $ 0$ or $ 2$, $ a_2$ is $ 0$ or $ 2$,...,$ a_{25}$ is $ 0$ or $ 2$? $ \textbf{(A)}\ 0 \le x < 1/3 \qquad \textbf{(B)}\ 1/3 \le x < 2/3 \qquad \textbf{(C)}\ 2/3 \le x < 1 \\ \textbf{(D)}\ 0 \le x < 1/3 \text{ or } 2/3 \le x < 1 \qquad \textbf{(E)}\ 1/2 \le x \le 3/4$

In rectangle $ABCD$, $AB=12$ and $BC=10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE=9$, $DF=8$, $\overline{BE} \parallel \overline{DF}$, $\overline{EF} \parallel \overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m\sqrt{n}-p$, where $m,n,$ and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$.

In regular pentagon $ABCDE$, let $O \in CE$ be the center of circle $\Gamma$ tangent to $DA$ and $DE$. $\Gamma$ meets $DE$ at $X$ and $DA$ at $Y$ . Let the altitude from $B$ meet $CD$ at $P$. If $CP = 1$, the area of $\vartriangle COY$ can be written in the form $\frac{a}{b} \frac{\sin c^o}{\cos^2 c^o}$ , where $a$ and $b$ are relatively prime positive integers and $c$ is an integer in the range $(0, 90)$. Find $a + b + c$.

$ABC$ is an acute triangle. $P$(different from $B,C$) is a point on side $BC$. $H$ is an orthocenter, and $D$ is a foot of perpendicular from $H$ to $AP$. The circumcircle of the triangle $ABD$ and $ACD$ is $O _1$ and $O_2$, respectively. A line $l$ parallel to $BC$ passes $D$ and meet $O_1$ and $O_2$ again at $X$ and $Y$, respectively. $l$ meets $AB$ at $E$, and $AC$ at $F$. Two lines $XB$ and $YC$ intersect at $Z$. Prove that $ZE=ZF$ is a necessary and sufficient condition for $BP=CP$.

Alice and Bob want to play a game. In the beginning of the game, they are teleported to two random position on a train, whose length is $ 1 $ km. This train is closed and dark. So they dont know where they are. Fortunately, both of them have iPhone 133, it displays some information: 1. Your facing direction 2. Your total walking distance 3. whether you are at the front of the train 4. whether you are at the end of the train Moreover, one may see the information of the other one. Once Alice and Bob meet, the game ends. Alice and Bob can only discuss their strategy before the game starts. Find the least value $ x $ so that they are guarantee to end the game with total walking distance $ \le x $ km.