Let circles $\omega_1$ and $\omega_2$ intersect at $P$ and $Q$. Let the line externally tangent to both circles that is closer to $Q$ touch $\omega_1$ at $A$ and $\omega_2$ at $B$. Let point $T$ lie on segment$ P Q$ such that $\angle AT B = 90^o$. Given that $AT = 6$, $BT = 8$, and $P T = 4$, compute $P Q$.

Prove the identity $ \left[ \frac{3+x}{6} \right] -\left[ \frac{4+x}{6} \right] +\left[ \frac{5+x}{6} \right] =\left[ \frac{1+x}{2} \right] -\left[ \frac{1+x}{3} \right] ,\quad\forall x\in\mathbb{R} , $ where $ [] $ is the integer part. [i]C. Mortici[/i]

Let $AA _1$ and $BB_1$ be the altitudes of an isosceles acute-angled triangle $ABC, M$ the midpoint of $AB$. The circles circumscribed around the triangles $AMA_1$ and $BMB_1$ intersect the lines $AC$ and $BC$ at points $K$ and $L$, respectively. Prove that $K, M$, and $L$ lie on the same line.

Inside the acute-angled triangle $ABC$ we take $P$ and $Q$ two isogonal conjugate points. The perpendicular lines on the interior angle-bisector of $\angle BAC$ passing through $P$ and $Q$ intersect the segments $AC$ and $AB$ at the points $B_p\in AC$, $B_q\in AC$, $C_p\in AB$ and $C_q\in AB$, respectively. Let $W$ be the midpoint of the arc $BAC$ of the circle $(ABC)$. The line $WP$ intersects the circle $(ABC)$ again at $P_1$ and the line $WQ$ intersects the circle $(ABC)$ again at $Q_1$. Prove that the points $P_1$, $Q_1$, $B_p$, $B_q$, $C_p$ and $C_q$ lie on a circle. [i]Proposed by P. Bibikov[/i]

Show that $x^4 + y^4 + z^2\ge xyz \sqrt8$ for all positive reals $x, y, z$.

The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number. .

Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is $2$ miles, which is $10,560$ feet, and Bella covers $2 \tfrac{1}{2}$ feet with each step. How many steps will Bella take by the time she meets Ella? $\textbf{(A) }704\qquad\textbf{(B) }845\qquad\textbf{(C) }1056\qquad\textbf{(D) }1760\qquad \textbf{(E) }3520$

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure? [img]https://1.bp.blogspot.com/-oWnh8uhyTIo/XzP30gZueKI/AAAAAAAAMUY/GlC3puNU_6g6YRf6hPpbQW8IE8IqMP3ugCLcBGAsYHQ/s0/2018%2BMohr%2Bp2.png[/img]

The minimum value of the quotient of a (base ten) number of three different nonzero digits divided by the sum of its digits is $\textbf{(A) }9.7\qquad\textbf{(B) }10.1\qquad\textbf{(C) }10.5\qquad\textbf{(D) }10.9\qquad \textbf{(E) }20.5$

Triangle I is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). Triangle II is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then: $ \textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad$ $\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad$ $\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes} $

Compute the number of ordered triples $(x,y,z)$ of integers satisfying \[x^2+y^2+z^2=9.\] [i]Proposed by Nathan Xiong[/i]

Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$. Prove that \[(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha\] if $0 \leq \alpha \leq 1$.

Fix positive integers $n$ and $k$ such that $2 \le k \le n$ and a set $M$ consisting of $n$ fruits. A [i]permutation[/i] is a sequence $x=(x_1,x_2,\ldots,x_n)$ such that $\{x_1,\ldots,x_n\}=M$. Ivan [i]prefers[/i] some (at least one) of these permutations. He realized that for every preferred permutation $x$, there exist $k$ indices $i_1 < i_2 < \ldots < i_k$ with the following property: for every $1 \le j < k$, if he swaps $x_{i_j}$ and $x_{i_{j+1}}$, he obtains another preferred permutation. \\ Prove that he prefers at least $k!$ permutations.

Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that \[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\] Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.

In the diagram below, fill the $12$ circles with numbers from the following bank so that each number is used once. Two circles connected by a single line must contain relatively prime numbers. Two circles connected by a double line must contain numbers that are not relatively prime. $$\text{Bank: } 20, 21, 22, 23, 24, 25, 27, 28, 30 ,32, 33 ,35$$ [asy] real HRT3 = sqrt(3) / 2; void drawCircle(real x, real y, real r) { path p = circle((x,y), r); draw(p); fill(p, white); } void drawCell(int gx, int gy) { real x = 0.5 * gx; real y = HRT3 * gy; drawCircle(x, y, 0.35); } void drawEdge(int gx1, int gy1, int gx2, int gy2, bool doubled) { real x1 = 0.5 * gx1; real y1 = HRT3 * gy1; real x2 = 0.5 * gx2; real y2 = HRT3 * gy2; if (doubled) { real dx = x2 - x1; real dy = y2 - y1; real ox = -0.035 * dy / sqrt(dx * dx + dy * dy); real oy = 0.035 * dx / sqrt(dx * dx + dy * dy); draw((x1+ox,y1+oy)--(x2+ox,y2+oy)); draw((x1-ox,y1-oy)--(x2-ox,y2-oy)); } else { draw((x1,y1)--(x2,y2)); } } drawEdge(2, 0, 4, 0, true); drawEdge(2, 0, 1, 1, true); drawEdge(2, 0, 3, 1, true); drawEdge(4, 0, 3, 1, false); drawEdge(4, 0, 5, 1, false); drawEdge(1, 1, 0, 2, false); drawEdge(1, 1, 2, 2, false); drawEdge(1, 1, 3, 1, false); drawEdge(3, 1, 2, 2, true); drawEdge(3, 1, 4, 2, true); drawEdge(3, 1, 5, 1, false); drawEdge(5, 1, 4, 2, true); drawEdge(5, 1, 6, 2, false); drawEdge(0, 2, 1, 3, false); drawEdge(0, 2, 2, 2, false); drawEdge(2, 2, 1, 3, false); drawEdge(2, 2, 3, 3, true); drawEdge(2, 2, 4, 2, false); drawEdge(4, 2, 3, 3, false); drawEdge(4, 2, 5, 3, false); drawEdge(4, 2, 6, 2, false); drawEdge(6, 2, 5, 3, true); drawEdge(1, 3, 3, 3, true); drawEdge(3, 3, 5, 3, false); drawCell(2, 0); drawCell(4, 0); drawCell(1, 1); drawCell(3, 1); drawCell(5, 1); drawCell(0, 2); drawCell(2, 2); drawCell(4, 2); drawCell(6, 2); drawCell(1, 3); drawCell(3, 3); drawCell(5, 3); [/asy]

A right cylinder is given with a height of $20$ and a circular base of radius $5$. A vertical planar cut is made into this base of radius $5$. A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the pieces is $100\sqrt2$. Then the volume of the larger piece can be written as $a + b\pi$, where $a, b$ are positive integers. Find $a + b$.

The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.

$ABC$ is a triangle with orthocenter $H$. The median from $A$ meets the circumcircle again at $A_1$, and $A_2$ is the reflection of $A_1$ in the midpoint of $BC$. The points$ B_2$ and $C_2$ are defined similarly. Show that $H$, $A_2$, $B_2$ and $C_2$ lie on a circle. [img]https://cdn.artofproblemsolving.com/attachments/f/1/192d14a0a7a9aa9ac7b38763e6ea6a4a95be55.png[/img]

There are integers $a$, $b$, and $c$, each greater than 1, such that $$\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}$$ for all $N > 1$. What is $b$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Initially, $10$ ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some $5$ numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by $1$. If in $10,000$ moves on the board a number divisible by $2023$ appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?

Write the equation of the line tangent to the graph of the function $y = x^4-x^2 + x$ to at least at two points.

Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown $a+b=c$ $d+e+f=g+h$ $i+j+k+l=m+n+o$ and so on... Where at the $j$-th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms. Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$. Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$. For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown. $1+3=4$ $2+5+8=6+9$ $7+11+12+14=13+15+16$

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$ $\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$