Triangle $ABC$ has $AB=4$, $BC=6$, and $AC=5$. Let $O$ denote the circumcenter of $ABC$. The circle $\Gamma$ is tangent to and surrounds the circumcircles of triangle $AOB$, $BOC$, and $AOC$. Determine the diameter of $\Gamma$.

(a) Sketch qualitativly the region with maximum area such that it lies in the first quadrant and is bound by $y=x^2-x^3$ and $y=kx$ where $k$ is a constent. The region must not have any other lines closing it. Note: $kx$ lies above $x^2-x^3$ (b) Find an expression for the volume of the solid obtained by spinning this region about the $y$ axis.

Point $P$ lies inside a triangle $ABC$. Let $D,E$ and $F$ be reflections of the point $P$ in the lines $BC,CA$ and $AB$, respectively. Prove that if the triangle $DEF$ is equilateral, then the lines $AD,BE$ and $CF$ intersect in a common point.

In MTRP district there are $10$ cities. Bob the builder wants to make roads between cities in such a way so that one can go from one city to the other through exactly one unique path. The government has allotted him a budget of Rs. $20$ and each road requires a positive integer amount (in Rs.) to build. In how many ways he can build such a network of roads? It is known that in the MTRP district, any positive integer amount of rupees can be used to construct a road, and that the full budget is used by Bob in the construction. Write the last two digits of your answer.

Let $N$ be an odd number, $N\geq 3$. $N$ tennis players take part in a championship. Before starting the championship, a commission puts the players in a row depending on how good they think the players are. During the championship, every player plays with every other player exactly once, and each match has a winner. A match is called [i]suprising[/i] if the winner was rated lower by the commission. At the end of the tournament, players are arranged in a line based on the number of victories they have achieved. In the event of a tie, the commission's initial order is used to decide which player will be higher. It turns out that the final order is exactly the same as the commission's initial order. What is the maximal number of suprising matches that could have happened.

Let $G$ be a connected (simple) graph with $n$ vertices and at least $n$ edges. Prove that it is possible to color the vertices of $G$ red and blue, so that the following conditions hold: i. There is at least one vertex of each color, ii. There is an even number of edges connecting a red vertex to a blue vertex, and iii. If all such edges are deleted, one is left with two connected graphs.

What is the $33$-rd number after the decimal point of $(\sqrt {10} + 3)^{2001}$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 8 $

Let a function $f$ satisfy $f(1) = 1$ and $f(1)+ f(2)+...+ f(n) = n^2f(n)$ for all $n \in N$. Determine $f(1995)$.

Let $f(0), f(1), f(2), \dots$ be a sequence satisfying \[ f(0) = 0 \quad \text{and} \quad f(n) = n - f(f(n-1)) \] for $n=1,2,3,\dots$. Give a formula for $f(n)$ such that its value can be immediately computed using $n$ without having to compute the previous terms.

If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form \[x=a_{1}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $a_{1}a_{2},\cdots$ are integers, $0 \le a_{n}\le n-1$ for $n>1$, and the series terminates. Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^{6}$.

In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s$ and $t$ is $1$. What is $r+s+t$? $\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 20$

The Fibonacci numbers $f_n$ are defined for each natural number $n$ as follows: $f_0=f_1=1$ and for $n$ greater than or equal to $2$, by recurrence: $f_n=f_{n-1}+f_{n-2}$ Let $S=f_1+f_2+...+f_{2004}+f_{2005}$. Calculate the largest value of $N$, such that the Fibonacci number $f_N$ satisfies $f_N<S$

[asy] size(120); path c = Circle((0, 0), 1); pair A = dir(20), B = dir(130), C = dir(240), D = dir(330); draw(c); pair F = 3(A-B) + B; pair G = 3(D-C) + C; pair E = intersectionpoints(B--F, C--G)[0]; draw(B--E--C--A); label("$A$", A, NE); label("$B$", B, NW); label("$C$", C, SW); label("$D$", D, SE); label("$E$", E, E); //Credit to MSTang for the diagram[/asy] In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$. $\textbf{(A) }10^\circ\qquad\textbf{(B) }15^\circ\qquad\textbf{(C) }20^\circ\qquad\textbf{(D) }\left(\frac{45}{2}\right)^\circ\qquad \textbf{(E) }30^\circ$

If $ a$, $ b$, $ x$, $ y$ are integers greater than 1 such that $ a$ and $ b$ have no common factor except 1 and $ x^a \equal{} y^b$ show that $ x \equal{} n^b$, $ y \equal{} n^a$ for some integer $ n$ greater than 1.

Find the smallest positive integer $n$ with the following property: there does not exist an arithmetic progression of $1999$ real numbers containing exactly $n$ integers.

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

Let $k$ and $N$ be integers such that $k > 1$ and $N > 2k + 1$. A number of $N$ persons sit around the Round Table, equally spaced. Each person is either a knight (always telling the truth) or a liar (who always lies). Each person sees the nearest k persons clockwise, and the nearest $k$ persons anticlockwise. Each person says: ''I see equally many knights to my left and to my right." Establish, in terms of $k$ and $N$, whether the persons around the Table are necessarily all knights. Sergey Berlov, Russia

[asy] size(180); defaultpen(linewidth(0.8)); real r=4/5; draw((-1,0)..(-6/7,r/3)..(0,r)..(6/7,r/3)..(1,0),linetype("4 4")); draw((-1,0)--(1,0)^^origin--(0,r)); label("$A$",(-1,0),W); label("$B$",(1,0),E); label("$M$",origin,S); label("$C$",(0,r),N); [/asy] A parabolic arch has a height of $16$ inches and a span of $40$ inches. The height, in inches, of the arch at a point $5$ inches from the center of $M$ is: $\textbf{(A) }1\qquad \textbf{(B) }15\qquad \textbf{(C) }15\tfrac13\qquad \textbf{(D) }15\tfrac12\qquad \textbf{(E) }15\tfrac34$

Find the number of solutions to the given congruence$$x^{2}+y^{2}+z^{2} \equiv 2 a x y z \pmod p$$ where $p$ is an odd prime and $x,y,z \in \mathbb{Z}$. [i]Proposed by math_and_me[/i]

Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$

There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels. The tournament works as follows: First, all $\binom{36}{2}$ pairings of players are written down on slips of paper and are placed in a hat. Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match. We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes. What is the expected value of the number of matches played before the stopping point is reached? [i]Proposed by Dilhan Salgado[/i]

Consider a regular polygon $A_1A_2\ldots A_{6n+3}$. The vertices $A_{2n+1}, A_{4n+2}, A_{6n+3}$ are called [i]holes[/i]. Initially there are three pebbles in some vertices of the polygon, which are also vertices of equilateral triangle. Players $A$ and $B$ take moves in turn. In each move, starting from $A$, the player chooses pebble and puts it to the next vertex clockwise (for example, $A_2\rightarrow A_3$, $A_{6n+3}\rightarrow A_1$). Player $A$ wins if at least two pebbles lie in holes after someone's move. Does player $A$ always have winning strategy? [i]Proposed by Bohdan Rublov [/i]

Given the number $\overline{a_1a_2\cdots a_n}$ such that $$\overline{a_n\cdots a_2a_1}\mid \overline{a_1a_2\cdots a_n}$$Then show $(\overline{a_1a_2\cdots a_n})(\overline{a_n\cdots a_2a_1})$ is a perfect square. [i]Proposed by Ezra Guerrero, Francisco Morazan[/i]

Find all real roots $x$ of the equation $$\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,$$ where $p$ is a real parameter.

A small point-like object is thrown horizontally off of a $50.0$-$\text{m}$ high building with an initial speed of $10.0 \text{ m/s}$. At any point along the trajectory there is an acceleration component tangential to the trajectory and an acceleration component perpendicular to the trajectory. How many seconds after the object is thrown is the tangential component of the acceleration of the object equal to twice the perpendicular component of the acceleration of the object? Ignore air resistance. $ \textbf{(A)}\ 2.00\text{ s}$ $\textbf{(B)}\ 1.50\text{ s}$ $\textbf{(C)}\ 1.00\text{ s}$ $\textbf{(D)}\ 0.50\text{ s}$ $\textbf{(E)}\ \text{The building is not high enough for this to occur.} $