Let $P$ and $Q$ be points in the interior of a triangle $ABC$ such that $\angle APC = \angle AQB = 90^{\circ}$, $\angle ACP = \angle PBC$, and $\angle ABQ = \angle QCB$. Suppose that lines $BP$ and $CQ$ meet at a point $R$. Show that $AR$ is perpendicular to $PQ$.

(A.Zaslavsky) The circumradius of triangle $ ABC$ is equal to $ R$. Another circle with the same radius passes through the orthocenter $ H$ of this triangle and intersect its circumcirle in points $ X$, $ Y$. Point $ Z$ is the fourth vertex of parallelogram $ CXZY$. Find the circumradius of triangle $ ABZ$.

For any partition $ P $ of $ [0,1] $ , consider the set $$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$ Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that $$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$ Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $ [0,1] . $

Given a rectangular table with 2 rows and 100 columns. Dima fills the cells of the first row with numbers 1, 2 or 3. Prove that Alex can fill the cells of the second row with numbers 1, 2, 3 in such a way that the numbers in each column are different and the sum of the numbers in the second row equals 200.

Given the set of numbers $ \{1, 2, 3, \ldots, 100\} $. From this set, create 10 pairwise disjoint subsets $ N_i = \{a_{i,1}, a_{i,2}, ... a_{i,10} $ ($ i = 1, 2, \ldots, 10 $ ) so that the sum of the products $$ \sum_{i=10}^{10}\prod_{j=1}^{10} a_{i,j} $$ was the biggest.

The non-parallel sides of a trapezium serve as the diameters of two circles. Prove that all four tangents to the circles drawn from the point of intersection of the diagonals are equal (if this point lies outside the circles). (S Markelov)

Let $c\neq 1$ be a positive rational number. Show that it is possible to partition $\mathbb{N}$, the set of positive integers, into two disjoint nonempty subsets $A,B$ so that $x/y\neq c$ holds whenever $x$ and $y$ lie both in $A$ or both in $B$.

Let $ a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0$ be real numbers. Show that \[ \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}). \] [i]Proposed by Romania[/i]

One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new rectangle. Find the length in feet of the perimeter of the new rectangle. [asy] size(250); defaultpen(linewidth(0.8)); draw((-2,0)--(-2,5)--(13,5)--(13,0)--cycle^^(16,-1)--(16,6)--(27,6)--(27,-1)--cycle^^(9,5)--(9,0)^^(16,4)--(27,4)); path rect1=(13,5)--(13,0)--(9,0)--(9,5)--cycle,rect2=(16,6)--(16,4)--(27,4)--(27,6)--cycle; fill(rect1,lightgray); fill(rect2,lightgray); draw(rect1^^rect2); [/asy]

Let $ABCD$ be a unit square. Points $E$ and $F$ are chosen on line segments $\overline{BC}$ and $\overline{CD}$, respectively, such that the area of $ABEF D$ is three times the area of triangle $\vartriangle ECF$. Compute the maximum possible area of triangle $\vartriangle AEF$.

Let $A_1, B_1, C_1$ be the feet of altitudes of an acute-angled triangle $ABC$. The incircle of triangle $A_1B_1C_1$ touches $A_1B_1, A_1C_1, B_1C_1$ at points $C_2, B_2, A_2$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ concur at a point lying on the Euler line of triangle $ABC$.

Let $a_1<a_2<\cdots<a_n$ be positive integers. Prove that $\displaystyle a_n \ge \sqrt[3]{\frac{(a_1+a_2+\cdots+a_n)^2}{n}}$.

Let $n$ be a positive integer such that no matter how $10^n$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0. Find the smallest possible value of $n$

Construct a right triangle, given two medians drawn to its legs.

Let the sequence an be defined by $a_0 = 2, a_1 = 15$, and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$. Show that there are infinitely many integers $k$ such that $269 | a_k$.

Find all differentiable functions $ f:(0,\infty )\longrightarrow (-\infty ,\infty ) $ having the property that $$ f'(\sqrt{x}) =\frac{1+x+x^2}{1+x} , $$ for any positive real numbers $ x. $ [i]Ovidiu Țâțan[/i]

Where is AMC 10a No.19? Thanks

On the table there are $N$ cards. Each card has an integer number written on it. Beto performs the following operation several times: he chooses two cards from the table, calculates the difference between the numbers written on them, writes the result on his notebook and removes those two cards from the table. He can perform this operation as many times as he wants, as long as there are at least two cards on the table. After this, Beto multiplies all the numbers that he wrote on his notebook. Beto's goal is that the result of this multiplication is a multiple of $7^{100}$. Find the minimum value of $N$ such that Beto can always achieve his goal, no matter what the numbers on the cards are.

A $20 \times 20$ checkered board is cut into several squares with integer side length. The size of a square is it's side length. What is the maximum amount of different sizes this squares can have?

Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line. [i]Proposed by Elliott Liu and Anthony Wang[/i]

Let $ABC$ be a triangle. Prove that there exists a unique point $P$ for which one can find points $D$, $E$ and $F$ such that the quadrilaterals $APBF$, $BPCD$, $CPAE$, $EPFA$, $FPDB$, and $DPEC$ are all parallelograms. [i]Proposed by Lewis Chen[/i]

Let $O$ be the circumcentre, and $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $P$ be an arbitrary point on $\Omega$, distinct from $A$, $B$, $C$, and their antipodes in $\Omega$. Denote the circumcentres of the triangles $AOP$, $BOP$, and $COP$ by $O_A$, $O_B$, and $O_C$, respectively. The lines $\ell_A$, $\ell_B$, $\ell_C$ perpendicular to $BC$, $CA$, and $AB$ pass through $O_A$, $O_B$, and $O_C$, respectively. Prove that the circumcircle of triangle formed by $\ell_A$, $\ell_B$, and $\ell_C$ is tangent to the line $OP$.

Rijul and Rohinee are playing a game on an $n\times n$ board alternating turns, with Rijul going first. In each turn, they fill an unfilled cell with a number from $1,2,\cdots, n^2$ such that no number is used twice. Rijul wins if there is any column such that the sum of all its elements is divisible by $n$. Rohinee wins otherwise. For what positive integers $n$ does he have a winning strategy? Proposed by Rohan Goyal

In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together? $\textbf{(A)} ~1\qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~24\qquad\textbf{(E)} ~120\qquad$

Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that \[8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , \] with equality if and only if triangle $ABC$ is equilateral.