Let $n\ge 4$ be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number $x$ in $[0, 1]$. At the start of the game, the only information Minivan has about $x$ is $x$ in $[0, 1]$. He needs to now learn about $x$ based on the following protocols: at each turn of his, Minivan chooses a number $y$ and submits to Megavan, where Megavan replies immediately with one of $y > x$, $y < x$, or $y\simeq x$, subject to two rules: $\bullet$ The answers in the form of $y > x$ and $y < x$ must be truthful; $\bullet$ Define the score of a round, known only to Megavan, as follows: $0$ if the answer is in the form $y > x$ and $y < x$, and $|x - y|$ if in the form $y\simeq x$. Then for every positive integer $k$ and every $k$ consecutive rounds, at least one round has score no more than $\frac{1}{k + 1}$. Minivan's goal is to produce numbers $a, b$ such that $a\le x\le b$ and $b - a\le \frac 1n$. Let $f(n)$ be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that $$n\le f(n) \le 4n$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

The area of a circle inscribed in a regular hexagon is $ 100\pi$. The area of hexagon is: $ \textbf{(A)}\ 600\qquad \textbf{(B)}\ 300\qquad \textbf{(C)}\ 200\sqrt{2}\qquad \textbf{(D)}\ 200\sqrt{3}\qquad \textbf{(E)}\ 120\sqrt{5}$

Given a triangle $ABC$, inside which the point $M$ is marked. On the sides $BC,CA$ and $AB$ the following points $A_1,B_1$ and $C_1$ are chosen, respectively, that $MA_1 \parallel CA$, $MB_1 \parallel AB$, $MC_1 \parallel BC$. Let S be the area of ​​triangle $ABC, Q_M$ be the area of ​​the triangle $A_1 B_1 C_1$. a) Prove that if the triangle $ABC$ is acute, and M is the point of intersection of its altitudes , then $3Q_M \le S$. Is there such a number $k> 0$ that for any acute-angled triangle $ABC$ and the point $M$ of intersection of its altitudes, such thatthe inequality $Q_M> k S$ holds? b) For cases where the point $M$ is the point of intersection of the medians, the center of the inscribed circle, the center of the circumcircle, find the largest $k_1> 0$ and the smallest $k_2> 0$ such that for an arbitrary triangle $ABC$, holds the inequality $k_1S \le Q_M\le k_2S$ (for the center of the circumscribed circle, only acute-angled triangles $ABC$ are considered).

How many of the numbers, $100,101,\ldots,999$, have three different digits in increasing order or in decreasing order? $\text{(A)} \ 120 \qquad \text{(B)} \ 168 \qquad \text{(C)} \ 204 \qquad \text{(D)} \ 216 \qquad \text{(E)} \ 240$

If the set $S = \{1,2,3,…,16\}$ is partitioned into $n$ subsets, there must be a subset in which elements $a, b, c$ (can be the same) exist, satisfying $a+ b=c$. Find the maximum value of $n$.

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.

Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as $$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.

If $ x^{2} \plus{}y^{2} \plus{}z\equal{}15$, $ x\plus{}y\plus{}z^{2} \equal{}27$ and $ xy\plus{}yz\plus{}zx\equal{}7$, then $\textbf{(A)}\ 3\le \left|x\plus{}y\plus{}z\right|\le 4 \\ \textbf{(B)}\ 5\le \left|x\plus{}y\plus{}z\right|\le 6 \\ \textbf{(C)}\ 7\le \left|x\plus{}y\plus{}z\right|\le 8 \\ \textbf{(D)}\ 9\le \left|x\plus{}y\plus{}z\right|\le 10 \\ \textbf{(E)}\ \text{None}$

A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.

Let $ABC$ be a triangle. $I$ is the incenter of $\Delta ABC$ and $D$ is the meet point of $AI$ and the circumcircle of $\Delta ABC$. Let $E,F$ be on $BD,CD$, respectively such that $IE,IF$ are perpendicular to $BD,CD$, respectively. If $IE+IF=\frac{AD}{2}$, find the value of $\angle BAC$.

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

Let $AA',BB', CC'$ be the bisectors of the angles of a triangle $ABC \ (A' \in BC, B' \in CA, C' \in AB)$. Prove that each of the lines $A'B', B'C', C'A'$ intersects the incircle in two points.

Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?

A polygon is [b]gridded[/b] if the internal angles of the polygon are either $90$ or $270$, it has integer side lengths and its sides don't intersect with each other. Prove that for all $n \ge 8$, it exist a gridded polygon with area $2n$ and perimeter $2n$.

Roy is starting a baking company and decides that he will sell cupcakes. He sells $n$ cupcakes for $(n + 20)(n + 15)$ cents. A man walks in and buys $\$10.50$ worth of cupcakes. Roy bakes cupcakes at a rate of $10$ cupcakes an hour. How many minutes will it take Roy to complete the order?

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

Find values of $n\in \mathbb{N}$ for which the fraction $\frac{3^n-2}{2^n-3}$ is reducible.

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds: $$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$

Solve the system of equations: \[x>y>z\] \[x+y+z=1\] \[x^2+y^2+z^2=69\] \[x^3+y^3+z^3=271\] [hide=Answer]x=7, y=-2, z=-4[/hide]

Among 16 coins there are 8 heavy coins with weight of 11 g, and 8 light coins with weight of 10 g, but it's unknown what weight of any coin is. One of the coins is anniversary. How to know, is anniversary coin heavy or light, via three weighings on scales with two cups and without any weight?

In a regular square room of side length $2\sqrt{2}$ ft, two cats that can see $2$ feet ahead of them are randomly placed into the four corners such that they do not share the same corner. If the probability that they don't see the mouse, also placed randomly into the room can be expressed as $\frac{a-b\pi}{c},$ where $a,b,c$ are positive integers with a greatest common factor of $1,$ then find $a+b+c.$ [i]Proposed by Ada Tsui[/i]

How many real solutions does the following system have ?$\begin{cases} x+y=2 \\ xy - z^2 = 1 \end{cases}$

Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is $ \text{(A)}\ 40\text{ dollars}\qquad\text{(B)}\ 50\text{ dollars}\qquad\text{(C)}\ 80\text{ dollars}\qquad\text{(D)}\ 100\text{ dollars}\qquad\text{(E)}\ 125\text{ dollars} $