Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

Let $a,b,c,d$ be real numbers such that \[a+b+c+d=-2\] \[ab+ac+ad+bc+bd+cd=0\] Prove that at least one of the numbers $a,b,c,d$ is not greater than $-1$.

Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? $ \textbf{(A)}\ 480 \qquad\textbf{(B)}\ 481 \qquad\textbf{(C)}\ 482 \qquad\textbf{(D)}\ 483 \qquad\textbf{(E)}\ 484 $

Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.

Given a permutation ($a_0, a_1, \ldots, a_n$) of the sequence $0, 1,\ldots, n$. A transportation of $a_i$ with $a_j$ is called legal if $a_i=0$ for $i>0$, and $a_{i-1}+1=a_j$. The permutation ($a_0, a_1, \ldots, a_n$) is called regular if after a number of legal transportations it becomes ($1,2, \ldots, n,0$). For which numbers $n$ is the permutation ($1, n, n-1, \ldots, 3, 2, 0$) regular?

Arrange arbitrarily $1,2,\ldots ,25$ on a circumference. We consider all $25$ sums obtained by adding $5$ consecutive numbers. If the number of distinct residues of those sums modulo $5$ is $d$ $(0\le d\le 5)$,find all possible values of $d$.

Let $a_1, a_2, ..., a_{1999}$ be nonnegative real numbers satisfying the following conditions: a. $a_1+a_2+...+a_{1999}=2$ b. $a_1a_2+a_2a_3+...+a_{1999}a_1=1$. Let $S=a_1^ 2+a_2 ^ 2+...+a_{1999}^2$. Find the maximum and minimum values of $S$.

In triangle $ABC, M$ is the midpoint of $AB$ and $D$ the foot of the bisector of angle $\angle ABC$. If $MD$ and $BD$ are known to be perpendicular, calculate $\frac{AB}{BC}$.

Let $ABCD$ and $WXYZ$ be two squares that share the same center such that $WX \parallel AB$ and $WX < AB.$ Lines $CX$ and $AB$ intersect at $P,$ and lines $CZ$ and $AD$ intersect at $Q.$ If points $P, W,$ and $Q$ are collinear, compute the ratio $AB/WX.$

Let $n, k, \ell$ be positive integers and $\sigma : \lbrace 1, 2, \ldots, n \rbrace \rightarrow \lbrace 1, 2, \ldots, n \rbrace$ an injection such that $\sigma(x)-x\in \lbrace k, -\ell \rbrace$ for all $x\in \lbrace 1, 2, \ldots, n \rbrace$. Prove that $k+\ell|n$.

Let $O$ be the center of the circle $\omega$, and let $A$ be a point inside this circle, different from $O$. Find all points $P$ on the circle $\omega$ for which the angle $\angle OPA$ acquires the greatest value.

Let $f : \mathbb{R} \to \mathbb{R}$ a periodic and continue function with period $T$ and $F : \mathbb{R} \to \mathbb{R}$ antiderivative of $f$. Then $$\int \limits_0^T \left[F(nx)-F(x)-f(x)\frac{(n-1)T}{2}\right]dx=0$$

Given a circle with its perimeter equal to $n$( $n \in N^*$), the least positive integer $P_n$ which satisfies the following condition is called the “[i]number of the partitioned circle[/i]”: there are $P_n$ points ($A_1,A_2, \ldots ,A_{P_n}$) on the circle; For any integer $m$ ($1\le m\le n-1$), there always exist two points $A_i,A_j$ ($1\le i,j\le P_n$), such that the length of arc $A_iA_j$ is equal to $m$. Furthermore, all arcs between every two adjacent points $A_i,A_{i+1}$ ($1\le i\le P_n$, $A_{p_n+1}=A_1$) form a sequence $T_n=(a_1,a_2,,,a_{p_n})$ called the “[i]sequence of the partitioned circle[/i]”. For example when $n=13$, the number of the partitioned circle $P_{13}$=4, the sequence of the partitioned circle $T_{13}=(1,3,2,7)$ or $(1,2,6,4)$. Determine the values of $P_{21}$ and $P_{31}$, and find a possible solution of $T_{21}$ and $T_{31}$ respectively.

Let $C_1$ and $C_2$ be two concentric reflective hollow metal spheres of radius $R$ and $R\sqrt3$ respectively. From a point $P$ on the surface of $C_2$, a ray of light is emitted inward at $30^o$ from the radial direction. The ray eventually returns to $P$. How many total reflections off of $C_1$ and $C_2$ does it take?

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

Define the function $f: \mathbb{N} \rightarrow \mathbb{N}$ which satisfies, for any $s, n \in \mathbb{N}$, the following conditions: $f(1) = f(2^s)$ and if $n < 2^s$, then $f(2^s + n) = f(n) + 1.$ Calculate the maximum value of $f(n)$ when $n \leq 2001$ and find the smallest natural number $n$ such that $f(n) = 2001.$

Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^2-2ax+b=0$ has real roots, given that $P(0)\cdot P(1)\cdot P(2)\neq 0$ and $P(0),P(1),P(2)$ form a geometric progression.

Find all $f:\mathbb{N}\to\mathbb{N}$ satisfying that for all $m,n\in\mathbb{N}$, the nonnegative integer $|f(m+n)-f(m)|$ is a divisor of $f(n)$. [i] Proposed by usjl[/i]

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$. [list=i] [*]For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$, solution of the equation $f(x)=x^n$. Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$. [*]Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$. For any $x\in\mathbb{R}$ we define \[F(x)=\int_x^1f(t) \ dt.\]Let $\alpha{}$ be a real number. Study the convergence of the series \[\sum_{n=1}^\infty F(a_n)^\alpha.\] [/list]

Find number of terms when we expand $(a+b+c)^{99}$ (in the general case).

At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

Let $ABC$ be a triangle and $\omega$ its incircle. $\omega$ touches $AC,AB$ at $E,F$, respectively. Let $P$ be a point on $EF$. Let $\omega_1=(BFP), \omega_2=(CEP)$. The parallel line through $P$ to $BC$ intersects $\omega_1,\omega_2$ at $X,Y$, respectively. Show that $BX=CY$.

Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is \[\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].\] Let those two collections satisfy the following properties: (i) On each row of $P$, from left to right, the numbers are non-increasing, (ii) On each column of $Q$, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in $P$ equals to the same row in $Q$, (iv) The sum of numbers on the column in $P$ equals to the same column in $Q$. Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$. [i]Proposer: Stefanus Lie[/i]

$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$, $az+cx=b$, $ay+bx=c$. Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$

Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.