Let $AH_a $ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ $ bisects segment $H_aH_b$. (A.Akopjan, K.Savenkov)

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

We call a polynomial $P(x)$ good if the numbers $P(k)$ and $P'(k)$ are integers for all integers $k$. Let $P(x)$ be a good polynomial of degree $d$, and let $N_d$ be the product of all composite numbers not exceeding $d$. Prove that the leading coefficient of the polynomial $N_d \cdot P(x)$ is integer.

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$

Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

The number of integral points satisfy $\begin{cases} y\leq 3x\\ y\geq \frac{x}{3}\\ x+y\geq100 \end{cases}$ on the coordinate plane is________.

Find all positive integers that can be written in the following way $\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}$ . Also, $a,b, c$ are positive integers that are pairwise relatively prime.

In isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\vartriangle ACM$ with the circle with center $B$ passing through $M$, such that $P \ne M$. If $\angle BPC = 135^o$, then $\frac{CP}{AP}$ can be written as $a +\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime. Find $a + b$.

We say that a digit is [i]high[/i] if it is placed between two other digits and it is bigger than both of them. The digits $0$,$1$,$2$,$\dots$,$9$ are used exactly once to form a 10-digit number. How many numbers can be formed with the property such that they don’t have any high digits?

Let $n \geq 2023$ be an integer. For each real $x$, we say that $\lfloor x \rceil$ is the closest integer to $x$, and if there are two closest integers then it is the greater of the two. Suppose there is a positive real $a$ such that $$\lfloor an \rceil = n + \bigg\lfloor\frac{n}{a} \bigg\rceil.$$ Show that $|a^2 - a - 1| < \frac{n\varphi+1}{n^2}$.

Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$. Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$.

There are $6$ computers(power off) and $3$ printers. Between a printer and a computer, they are connected with a wire or not. Printer can be only activated if and only if at least one of the connected computer's power is on. Your goal is to connect wires in such a way that, no matter how you choose three computers to turn on among the six, you can activate all $3$ printers. What is the minimum number of wires required to make this possible?

A point is "bouncing" inside a unit equilateral triangle with vertices $(0,0)$, $(1,0)$, and $(1/2,\sqrt{3}/2)$. The point moves in straight lines inside the triangle and bounces elastically off an edge at an angle equal to the angle of incidence. Suppose that the point starts at the origin and begins motion in the direction of $(1,1)$. After the ball has traveled a cumulative distance of $30\sqrt{2}$, compute its distance from the origin.

Inside the equilateral triangle $ABC$ lies the point $T$. Prove that $TA$, $TB$ and $TC$ are the lengths of the sides of a triangle.

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Let the angle bisectors of $\angle BAC,$ $\angle CBA,$ and $\angle ACB$ meets the circumcircle of $\triangle ABC$ at the points $M,N,$ and $K,$ respectively. Let the segments $AB$ and $MK$ intersects at the point $P$ and the segments $AC$ and $MN$ intersects at the point $Q.$ Prove that $PQ\parallel BC$

In a triangle $ABC$ a circle $k$ is inscribed, which is tangent to $BC$,$CA$,$AB$ in points $D,E,F$ respectively. Let point $P$ be inner for $k$. If the lines $DP$,$EP$,$FP$ intersect $k$ in points $D',E',F'$ respectively, then prove that $AD'$, $BE'$, and $CF'$ are concurrent.

A number $N$ has 2009 positive factors. What is the maximum number of positive factors that $N^2$ could have?

Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.

Two players play a game with a pile of $N$ coins on a table. On a player's turn, if there are $n$ coins, the player can take at most $n/2+1$ coins, and must take at least one coin. The player who grabs the last coin wins. For how many values of $N$ between $1$ and $100$ (inclusive) does the first player have a winning strategy?

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

Prove that however we choose the majority of numbers among an even number of the first consecutive natural numbers, there will be two numbers among this choosing whose sum is a prime.

The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$. [asy] path seg1, seg2; seg1=(6,0)--(0,3); seg2=(2,0)--(4,2); dot((0,0)); dot((1,0)); fill(circle((2,0),0.1),black); dot((3,0)); dot((4,0)); dot((5,0)); fill(circle((6,0),0.1),black); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((4,1)); dot((5,1)); dot((6,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((3,2)); fill(circle((4,2),0.1),black); dot((5,2)); dot((6,2)); fill(circle((0,3),0.1),black); dot((1,3)); dot((2,3)); dot((3,3)); dot((4,3)); dot((5,3)); dot((6,3)); draw(seg1); draw(seg2); pair [] x=intersectionpoints(seg1,seg2); fill(circle(x[0],0.1),black); label("$A$",(0,3),NW); label("$B$",(6,0),SE); label("$C$",(4,2),NE); label("$D$",(2,0),S); label("$E$",x[0],N);[/asy] $\text{(A)}\ \frac{4\sqrt5}{3}\qquad\text{(B)}\ \frac{5\sqrt5}{3}\qquad\text{(C)}\ \frac{12\sqrt5}{7}\qquad\text{(D)}\ 2\sqrt5 \qquad\text{(E)}\ \frac{5\sqrt{65}}{9}$

Let $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$, where $n$ is an integer greater than $1$. Find the smallest number $k=k(n)$ such that there is a number which may be expressed as a sum of $k$ (possibly repeated) elements in $S_{n}$ in more than one way. (Rearrangements are considered the same.)

Find all positive integers $n$ such that there exists an infinite set $A$ of positive integers with the following property: For all pairwise distinct numbers $a_1, a_2, \ldots , a_n \in A$, the numbers $$a_1 + a_2 + \ldots + a_n \text{ and } a_1\cdot a_2\cdot \ldots\cdot a_n$$ are coprime.