Mr. Squash bought a large parking lot in Utah, which has an area of $600$ square meters. A car needs $6$ square meters of parking space while a bus needs $30$ square meters of parking space. Mr. Squash charges $\$2.50$ per car and $\$7.50$ per bus, but Mr. Squash can only handle at most $60$ vehicles at a time. Find the ordered pair $(a,b)$ where $a$ is the number of cars and $b$ is the number of buses that maximizes the amount of money Mr. Squash makes. [i]Proposed by Nathan Cho[/i]

Let $\Delta ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$. Contruct the circle with diameter $\overline{BC}$, and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$, respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$. What is the degree measure of $\angle BFC ?$ $\textbf{(A) } 90 \qquad\textbf{(B) } 100 \qquad\textbf{(C) } 105 \qquad\textbf{(D) } 110 \qquad\textbf{(E) } 120$

A parking lot has $ 16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $ 2$ adjacent spaces. What is the probability that she is able to park? $ \textbf{(A)} \ \frac {11}{20} \qquad \textbf{(B)} \ \frac {4}{7} \qquad \textbf{(C)} \ \frac {81}{140} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {17}{28}$

Find all pair of positive integers $(m,n)$ such that $$mn(m^2+6mn+n^2)$$is a perfect square. [i]Proposed by Li4 and Untro368[/i]

Let $a,b,c$ be positive integers such that $a$ divides $b^2$, $b$ divides $c^2$ and $c$ divides $a^2$. Prove that $abc$ divides $(a + b +c)^7$.

For all positive real numbers $a,b,c$ with $a+b+c=3$, show that $$a^3b+b^3c+c^3a+9\geq 4(ab+bc+ca).$$

Let $a,b,c$ be the sides and $\alpha,\beta,\gamma$ be the corresponding angles of a triangle. Prove the equality $$\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.$$

Let $2561$ given points on a circle be colored either red or green. In each step, all points are recolored simultaneously in the following way: if both direct neighbors of a point $P$ have the same color as $P$, then the color of $P$ remains unchanged, otherwise $P$ obtains the other color. Starting with the initial coloring $F_1$, we obtain the colorings $F_2, F_3,\dots$ after several recoloring steps. Determine the smallest number $n$ such that, for any initial coloring $F_1$, we must have $F_n = F_{n+2}$.

Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that: $$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$

Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively. $(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$. $(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .

Let $C_1$ and $C_2$ be two circumferences externally tangents at $S$ such that the radius of $C_2$ is the triple of the radius of $C_1$. Let a line be tangent to $C_1$ at $P \neq S$ and to $C_2$ at $Q \neq S$. Let $T$ be a point on $C_2$ such that $QT$ is diameter of $C_2$. Let the angle bisector of $\angle SQT$ meet $ST$ at $R$. Prove that $QR=RT$

Let an acute-angled triangle $ABC$ with $AB<AC<BC$, inscribed in circle $c(O,R)$. The angle bisector $AD$ meets $c(O,R)$ at $K$. The circle $c_1(O_1,R_1)$(which passes from $A,D$ and has its center $O_1$ on $OA$) meets $AB$ at $E$ and $AC$ at $Z$. If $M,N$ are the midpoints of $ZC$ and $BE$ respectively, prove that: [b]a)[/b]the lines $ZE,DM,KC$ are concurrent at one point $T$. [b]b)[/b]the lines $ZE,DN,KB$ are concurrent at one point $X$. [b]c)[/b]$OK$ is the perpendicular bisector of $TX$.

For a positive integer $n$, a [i]sum-friendly odd partition[/i] of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be [b]uniquely[/b] written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.

Which of the following polynomials has the greatest real root? $\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018 $

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

A series of numbers is called complete if it has non-zero natural terms and any nonzero integer has at least one among multiple series. Show that the arithmetic progression is a complete sequence if and only if it divides the first term relationship.

Prove that after completing the multiplication and collecting the terms $$(1 - x + x^2 - x^3 +... - x^{99} + x^{100})(1 + x + x^2 + ...+ x^{99} + x^{100})$$ has no monomials of odd degree.

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

Call a number [i]feared [/i] if it contains the digits $13$ as a contiguous substring and [i]fearless [/i] otherwise. (For example, $132$ is feared, while $123$ is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a < 100$ such that $n$ and $n + 10a$ are fearless while $n +a$, $n + 2a$, $. . . $, $n + 9a$ are all feared.

On the surface of a sphere, a non-intersecting closed curve is drawn. It divides the surface of the sphere in two regions, which are coloured red and blue. Prove that there exist two antipodes of different colours. [i]Note: the curve is colourless.[/i] [i]Proposed by Vlad Spătaru[/i]

Define an $n$-digit pair cycle to be a number with $n^2 + 1$ digits between $1$ and $n$ with every possible pair of consecutive digits. For instance, $11221$ is a 2-digit pair cycle since it contains the consecutive digits $11$, $12$, $22$, and $21$. How many $3$-digit pair cycles exist?

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect the circumcircle of this triangle at points $A_0$ and $C_0$, respectively. A straight line passing through the center of the inscribed circle of a triangle $ABC$ is parallel to side $AC$ and intersects line $A_0C_0$ at point $P$. Prove that line $PB$ is tangent to the circumcircle of the triangle $ABC$.

Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.

Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]