There are $n$ cards. Max and Lewis play, alternately, the following game Max starts the game, he removes exactly $1$ card, in each round the current player can remove any quantity of cards, from $1$ card to $t+1$ cards, which $t$ is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of $n$ such that Max has the winning strategy.

In a triangle $ABC$ with $AC\ne BC$, $M$ is the midpoint of $AB$ and $\angle A=\alpha$, $\angle B=\beta$, $\angle ACM=\varphi$ and $\angle BSM=\Psi$. Prove that $$\frac{\sin\alpha\sin\beta}{\sin(\alpha-\beta)}=\frac{\sin\varphi\sin\Psi}{\sin(\varphi-\Psi)}.$$

Let $\triangle ABC$ be a triangle with $BA<AC$, $BC=10$, and $BA=8$. Let $H$ be the orthocenter of $\triangle ABC$. Let $F$ be the point on segment $AC$ such that $BF=8$. Let $T$ be the point of intersection of $FH$ and the extension of line $BC$. Suppose that $BT=8$. Find the area of $\triangle ABC$.

What is the maximal number of solutions can the equation have $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$ where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$.

$m(m>1)$ is an intenger, define $(a_n)$: $a_0=m,a_{n}=\varphi(a_{n-1})$ for all positive intenger $n$. If for all nonnegative intenger $k$, $a_{k+1}\mid a_k$, find all $m$ that is not larger than $2016$. Note: $\varphi(n)$ means Euler Function.

Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let $R$ be the region in which two circles overlap. Determine the perimeter of $R.$

[b] Newton's Law of Gravity from Kepler's Laws?[/b] [list=1] [*] Planets in the solar system move in elliptic orbits with the sun at one of the foci. [/*] [*] The line joining the sun and the planet sweeps out equal areas in equal times. [/*] [*] The period of revolution ($T$) and the length of the semi-major axis $(a$) of the ellipse sit in the relation $T^2/a^3 = constant$. [/*] [/list] Now answer the following questions: [list] [*] Starting from Newton's Law of Gravitation and Kepler's first law, derive the second and third law. It is possible to derive the first law but that is beyond the scope of this exam. [/*] [*] For convenience work in the complex (Argand) plane and take the sun to be at the origin $(z=0)$. Show that points on the ellipse may be represented by, \[ z(\theta) = \frac{a(1-\epsilon^2)}{1+\epsilon\cos\theta}\exp(i\theta) = r(\theta) e^{i\theta}\] where $a$ is the length of the semi-major axis, $\epsilon$ is the eccentricity of the ellipse and $\theta$ is called the \emph{true anomaly} in celestial mechanics. [/*] [*] Show that Kepler's second law leads to, \[ \frac{1}{2}r^2 \dot{\theta} = constant\] where $r$ and $\theta$ are defined as in part (b) and a dot $(.)$ over a variable denotes its time derivative. What is this constant in terms of the other variables of the problem? [/*] [*] Using the results of parts (b) and (c) along with Kepler's third law obtain Newton's Law of Gravitation. [/*] [*] Can the above exercise truly be called a "derivation" of Newton's Law of Gravitation? State your reasons. [/*] [/list]

Let $m$ and $n$ be odd positive integers. Each square of an $m$ by $n$ board is coloured red or blue. A row is said to be red-dominated if there are more red squares than blue squares in the row. A column is said to be blue-dominated if there are more blue squares than red squares in the column. Determine the maximum possible value of the number of red-dominated rows plus the number of blue-dominated columns. Express your answer in terms of $m$ and $n$.

In quadrilateral $ABCD$ , $AC$ is bisector of $\hat{A}$ and $\widehat{ADC}=\widehat{ACB}$. $X$ and $Y$ are feet of perpendicular from $A$ to $BC$ and $CD$,respectively.Prove that orthocenter of triangle $AXY$ is on $BD$.

In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$,$\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$. Then find the angle $\angle{DBA}$

* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.

Solve the following equation in $\mathbb{R}$: $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$

Through a point $P$ inside the triangle $ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas. If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: $ \textbf{(A)}\ \frac12 \qquad\textbf{(B)}\ \frac14\qquad\textbf{(C)}\ 2-\sqrt2\qquad\textbf{(D)}\ \frac{2-\sqrt2}{2}\qquad\textbf{(E)}\ \frac{2+\sqrt2}{8} $

In the cyclic quadrilateral $ABCD$ with $AB = AD$, points $M$ and $N$ lie on the sides $CD$ and $BC$ respectively so that $MN = BN + DM$. Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$.

In the expansion of $ (a \plus{} b)^n$ there are $ n \plus{} 1$ dissimilar terms. The number of dissimilar terms in the expansion of $ (a \plus{} b \plus{} c)^{10}$ is: $ \textbf{(A)}\ 11\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 55\qquad \textbf{(D)}\ 66\qquad \textbf{(E)}\ 132$

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

A freight train leaves the town of Jenkinsville at $1:00$ PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at $10$ miles per hour. At $1:12$ PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at $3:5$ miles per hour directly towards the road on which Julie is riding. Jim reaches the road at $2:12$ PM, just as Julie is riding by. What is the speed of the train in miles per hour?

Let ${a_1,a_2,\dots,a_n}$ be positive real numbers, ${n>1}$. Denote by $g_n$ their geometric mean, and by $A_1,A_2,\dots,A_n$ the sequence of arithmetic means defined by \[ A_k=\frac{a_1+a_2+\cdots+a_k}{k},\qquad k=1,2,\dots,n. \] Let $G_n$ be the geometric mean of $A_1,A_2,\dots,A_n$. Prove the inequality \[ n \root n\of{\frac{G_n}{A_n}}+ \frac{g_n}{G_n}\le n+1 \] and establish the cases of equality. [i]Proposed by Finbarr Holland, Ireland[/i]

Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal $AC$ does not exceed 1500. [i]Proposed by A. Kanel-Belov[/i]

Let $a,b,c$ be real nonzero numbers such that $a+b+c=12$ and \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}=1.\] Compute the largest possible value of $abc-\left(a+2b-3c\right)$. [i]Proposed by Tristan Shin[/i]

Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.

$(\sqrt6 + \sqrt7)^{1000}$ in base ten has a tens digit of $a$ and a ones digit of $b$. Determine $10a + b$.

A cylinder with radius $15$ and height $16$ is inscribed in a sphere. Three congruent smaller spheres of radius $x$ are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of $x$?

Three non-negative integers are written on a blackboard. A move is to replace two of the integers $k,m$ by $k+m$ and $|k-m|$. Determine whether we can always end with triplet which has at least two zeros

There are $1000$ rooms in a row along a long corridor. Initially the first room contains $1000$ people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?