Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.

A line initially $ 1$ inch long grows according to the following law, where the first term is the initial length. \[ 1 \plus{} \frac {1}{4}\sqrt {2} \plus{} \frac {1}{4} \plus{} \frac {1}{16}\sqrt {2} \plus{} \frac {1}{16} \plus{} \frac {1}{64}\sqrt {2} \plus{} \frac {1}{64} \plus{} \cdots. \]If the growth process continues forever, the limit of the length of the line is: $ \textbf{(A)}\ \infty \qquad\textbf{(B)}\ \frac {4}{3} \qquad\textbf{(C)}\ \frac {8}{3} \qquad\textbf{(D)}\ \frac {1}{3}(4 \plus{} \sqrt {2}) \qquad\textbf{(E)}\ \frac {2}{3}(4 \plus{} \sqrt {2})$

The word "'''HELP'''" in block letters is painted in black with strokes $1$ unit wide on a $5$ by $15$ rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is [asy] unitsize(12); fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black); fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black); fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black); fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black); fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white); draw((0,0)--(15,0)--(15,5)--(0,5)--cycle); label("$5\left\{ \begin{tabular}{c} \\ \\ \\ \\ \end{tabular}\right.$",(1,2.5),W); label(rotate(90)*"$\{$",(0.5,0.1),S); label("$1$",(0.5,-0.6),S); label(rotate(90)*"$\{$",(3.5,0.1),S); label("$1$",(3.5,-0.6),S); label(rotate(90)*"$\{$",(7.5,0.1),S); label("$1$",(7.5,-0.6),S); label(rotate(90)*"$\{$",(11.5,0.1),S); label("$1$",(11.5,-0.6),S); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(1.5,4),N); label("$3$",(1.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(5.5,4),N); label("$3$",(5.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(9.5,4),N); label("$3$",(9.5,5.8),N); label(rotate(270)*"$\left\{ \begin{tabular}{c} \\ \\ \end{tabular}\right.$",(13.5,4),N); label("$3$",(13.5,5.8),N); label("$\left. \begin{tabular}{c} \\ \end{tabular}\right\} 2$",(14,1),E); [/asy] $\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 34 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 38$

Given a positive real number $t$, find the number of real solutions $a, b, c, d$ of the system \[a(1 - b^2) = b(1 -c^2) = c(1 -d^2) = d(1 - a^2) = t.\]

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

For triangle $\vartriangle ABC$, define its $A$-excircle to be the circle that is externally tangent to line segment $BC$ and extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, and define the $B$-excircle and $C$-excircle likewise. Then, define the $A$-[i]veryexcircle [/i] to be the unique circle externally tangent to both the $A$-excircle as well as the extensions of $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$, but that shares no points with line $\overleftrightarrow{BC}$, and define the $B$-veryexcircle and $C$-veryexcircle likewise. Compute the smallest integer $N \ge 337$ such that for all $N_1 \ge N$, the area of a triangle with lengths $3N^2_1$ , $3N^2_1 + 1$, and $2022N_1$ is at most $\frac{1}{22022}$ times the area of the triangle formed by connecting the centers of its three veryexcircles. If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^3\right\rfloor \right)$.

Let $ABC$ be a triangle, and $P$ be a variable point inside $ABC$ such that $AP$ and $CP$ intersect sides $BC$ and $AB$ at $D$ and $E$ respectively, and the area of the triangle $APC$ is equal to the area of quadrilateral $BDPE$. Prove that the circumscribed circumference of triangle $BDE$ passes through a fixed point different from $B$.

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

In triangle $ABC$ the midpoints of sides $AC, BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A, B$ and the orthocenter of triangle $ABC$.

George has $150$ cups of flour and $200$ eggs. He can make a cupcake with $3$ cups of flour and $2$ eggs, or he can make an omelet with $4$ eggs. What is the maximum number of treats (both omelets and cupcakes) he canmake?

Inside the acute-angled triangle $ABC$, the points $P{}$ and $Q{}$ are chosen so that $\angle ACP = \angle BCQ$ and $\angle CBP =\angle ABQ$. The point $Z{}$ is the projection of $P{}$ onto the line $BC$. The point $Q'$ is symmetric to $Q{}$ with respect to $Z{}$. The points $K{}$ and $L{}$ are chosen on the rays $AB$ and $AC$ respectively, so that $Q'K \parallel QC$ and $Q'L \parallel QB$. Prove that $\angle KPL=\angle BPC$.

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+y) + f(x)f(y) = f(xy) + (y+1)f(x) + (x+1)f(y)\] for all $x,y \in \mathbb{R}$.

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

A subset $S \in \{0, 1, 2, \cdots , 2000\}$ satisfies $|S|=401$. Prove that there exists a positive integer $n$ such that there are at least $70$ positive integers $x$ such that $x, x+n \in S$

Let $a > b > c > d$ be positive integers and suppose that \[ ac + bd = (b+d+a-c)(b+d-a+c). \] Prove that $ab + cd$ is not prime.

Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? $\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$

Let $a, b, c$ be positive real numbers such that $abc = \dfrac {1} {8}$. Prove the inequality:$$a ^ 2 + b ^ 2 + c ^ 2 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2a ^ 2 \geq \dfrac {15} {16}$$ When the equality holds?

prove that $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) \leq 1$ for $0 < a < b \leq c < d$ and when $(\frac{1}{a+c}+\frac{1}{b+d})(\frac{1}{\frac{1}{a}+\frac{1}{c}}+\frac{1}{\frac{1}{b}+\frac{1}{d}}) = 1 $

Solve the equation \[(a^{2},b^{2})+(a,bc)+(b,ac)+(c,ab)=199.\] in positive integers. (Here $(x,y)$ denotes the greatest common divisor of $x$ and $y$.) [i]Proposed by S. Berlov[/i]

Victor shuffles a standard 54-card deck then flips over cards one at a time onto a pile stopping after the first ace. However, if he ever reveals a joker he discards the entire pile, including the joker, and starts a new pile; for example, if the sequence of cards is 2-3-Joker-A, the pile ends with one card in it. Find the expected number of cards in the end pile.

Let $a,b,$ and $c$ be real numbers such that \begin{align*} a+b+c &= 100 \\ ab+bc+ca &= 20, \text{ and} \\ (a+b)(a+c) &=24. \end{align*} Compute all possible values of $bc.$

It is desired to choose $n$ integers from the collection of $2n$ integers, namely, $0,0,1,1,2,2,\ldots,n-1,n-1$ such that the average of these $n$ chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer $n$ and find this minimum value for each $n$.

Prove that for all sufficiently large positive integers $d{}$, at least $99\%$ of the polynomials of the form \[\sum_{i\leqslant d}\sum_{j\leqslant d}\pm x^iy^j\]are irreducible over the integers.

Let $S$ denote the set of rational numbers different from $ \{ -1, 0, 1 \} $. Define $f: S \rightarrow S $ by $f(x)=x-1/x$. Prove or disprove that \[ \cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset \] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.

The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country.