A simplified model of a bicycle of mass $M$ has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is $w$, and the center of mass of the bicycle is located midway between the tires and a height h above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude $a$. Air resistance may be ignored. [asy] size(175); pen dps = linewidth(0.7) + fontsize(4); defaultpen(dps); draw(circle((0,0),1),black+linewidth(2.5)); draw(circle((3,0),1),black+linewidth(2.5)); draw((1.5,0)--(0,0)--(1,1.5)--(2.5,1.5)--(1.5,0)--(1,1.5),black+linewidth(1)); draw((3,0)--(2.4,1.8),black+linewidth(1)); filldraw(circle((1.5,2/3),0.05),gray); draw((1.3,1.6)--(0.7,1.6)--(0.7,1.75)--cycle,black+linewidth(1)); label("center of mass of bicycle",(2.5,1.9)); draw((1.55,0.85)--(1.8,1.8),BeginArrow); draw((4.5,-1)--(4.5,2/3),BeginArrow,EndArrow); label("$h$",(4.5,-1/6),E); draw((1.5,2/3)--(4.5,2/3),dotted); draw((0,-1)--(4.5,-1),dotted); draw((0,-5/4)--(3,-5/4),BeginArrow,EndArrow); label("$w$",(3/2,-5/4),S); draw((0,-1)--(0,-6/4),dotted); draw((3,-1)--(3,-6/4),dotted); [/asy] Case 1 ([b][u]Questions 28 - 29[/u][/b]): Assume that the coefficient of sliding friction between each tire and the ground is $\mu$, and that both tires are skidding: sliding without rotating. Express your answers in terms of $w$, $h$, $M$, and $g$. What is the maximum value of $a$ so that both tires remain in contact with the ground? $ \textbf{(A)}\ \frac{wg}{h} $ $ \textbf{(B)}\ \frac{wg}{2h}$ $ \textbf{(C)}\ \frac{hg}{2w} $ $ \textbf{(D)}\ \frac{h}{2wg} $ $ \textbf{(E)}\ \text{none of the above} $

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

[5] Find all integers $n$ for which $\frac{n^3+8}{n^2-4}$ is an integer.

Do there exist two positive reals $\alpha, \beta$ such that each positive integer appears exactly once in the following sequence? \[ 2020, [\alpha], [\beta], 4040, [2\alpha], [2\beta], 6060, [3\alpha], [3\beta], \cdots \] If so, determine all such pairs; if not, prove that it is impossible.

Prove that if the roots of the equation $ x^3 + ax^2 + bx + c = 0 $, with real coefficients, are real, then the roots of the equation $ 3x^2 + 2ax + b = 0 $ are also real.

Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

The circles $C_1, C_2$ and $C_3$ are externally tangent in pairs (each tangent to other two externally). Let $M$ the common point of $C_1$ and $C_2, N$ the common point of $C_2$ and $C_3$ and $P$ the common point of $C_3$ and $C_1$. Let $A$ be an arbitrary point of $C_1$. Line $AM$ cuts $C_2$ in $B$, line $BN$ cuts $C_3$ in $C$ and line $CP$ cuts $C_1$ in $D$. Prove that $AD$ is diameter of $C_1$.

Let $I$ be the center of incircle of $\triangle ABC$, and $J$ be the center of excircle tangent to $[BC]$. If $m(\widehat B) = 45^\circ$, $m(\widehat A) = 120^\circ$, and $|IJ|=\sqrt 3$, then what is $|BC|$? $ \textbf{(A)}\ \frac 32 \qquad\textbf{(B)}\ \frac {\sqrt 3}2 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac {\sqrt 6}2 \qquad\textbf{(E)}\ \sqrt3 - 1 $

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

Prove that for every integer $n > 2$, $$(1\cdot 2 \cdot 3 \cdot ... \cdot n)^2 > n^n.$$

Let $n$ be a positive integer. If $S$ is a nonempty set of positive integers, then we say $S$ is $n$-[i]complete [/i] if all elements of $S$ are divisors of $n$, and if $d_1$ and $d_2$ are any elements of $S$, then $n| d_1$ and gcd $(d_1, d_2)$ are in $S$. How many $2310$-complete sets are there?

Let $ABC$ be a triangle and $D$ is a point inside of $\triangle ABC$. The point $A'$ is the midpoint of the arc $BDC$, in the circle which passes by $B,C,D$. Analogously define $B'$ and $C'$, being the midpoints of the arc $ADC$ and $ADB$ respectively. Prove that the four points $D,A',B',C'$ are concyclic.

Sequence $(a_n)$ is defined as $a_{n+1}-2a_n+a_{n-1}=7$ for every $n\geq 2$, where $a_1 = 1, a_2=5$. What is $a_{17}$? $ \textbf{(A)}\ 895 \qquad\textbf{(B)}\ 900 \qquad\textbf{(C)}\ 905 \qquad\textbf{(D)}\ 910 \qquad\textbf{(E)}\ \text{None of the above} $

There is a knight in the left down corner of $2009 \times 2009$ chessboard. The row and the column containing this corner are painted. The knight cannot move into painted cell and after its move new row and column that contains a square with knight become painted. Is it possible to paint all rows and columns of the chessboard?

Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible? $\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$

Can the set of lattice points $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \le x, y \le 252, x \neq y\}$ be colored using 10 distinct colors such that for all $a \neq b$, $b \neq c$, the colors of $(a, b)$ and $(b, c)$ are distinct?

Prove that : $\cos36-\sin18=\frac{1}{2}$

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.

Let $ABC$ be a triangle and $AA_1$ be the angle bisector of $A$ ($A_1 \in BC$). The point $P$ is on the segment $AA_1$ and $M$ is the midpoint of the side $BC$. The point $Q$ is on the line connecting $P$ and $M$ such that $M$ is the midpoint of $PQ$. Define $D$ and $E$ as the intersections of $BQ$, $AC$, and $CQ$, $AB$. Prove that $CD=BE$.

For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.

Prove that there exist two strictly increasing sequences $a_{n}$ and $b_{n}$ such that $a_{n}(a_{n} +1)$ divides $b_{n}^2 +1$ for every natural $n$.

The [i]liar's guessing game[/i] is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game $A$ chooses integers $x$ and $N$ with $1 \le x \le N.$ Player $A$ keeps $x$ secret, and truthfully tells $N$ to player $B$. Player $B$ now tries to obtain information about $x$ by asking player $A$ questions as follows: each question consists of $B$ specifying an arbitrary set $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$. Player $B$ may ask as many questions as he wishes. After each question, player $A$ must immediately answer it with [i]yes[/i] or [i]no[/i], but is allowed to lie as many times as she wants; the only restriction is that, among any $k+1$ consecutive answers, at least one answer must be truthful. After $B$ has asked as many questions as he wants, he must specify a set $X$ of at most $n$ positive integers. If $x$ belongs to $X$, then $B$ wins; otherwise, he loses. Prove that: 1. If $n \ge 2^k,$ then $B$ can guarantee a win. 2. For all sufficiently large $k$, there exists an integer $n \ge (1.99)^k$ such that $B$ cannot guarantee a win. [i]Proposed by David Arthur, Canada[/i]