Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

In the grid to the right, the shortest path through unit squares between between the pair of 2's has length 2. Fill in some of the unit squares in the grid so that (i) exactly half of the squares in each row and column contain a number, (ii) each of the number 1 through 12 appears exactly twice, and (iii) for $n=1,2,\cdot\cdot\cdot,12$, the shortest path between the pair of $n$'s has length exactly $n$.

What is the largest integer less than or equal to $$\frac{3^{31}+2^{31}}{3^{29}+2^{29}} \,\,\, ?$$

Construct a convex polyhedron of equal “bricks” shown in Figure. [img]https://cdn.artofproblemsolving.com/attachments/6/6/75681a90478f978665b6874d0c0c9441ea3bd2.gif[/img]

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and \[\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10\]

Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?

$ABCD$ is a cyclic quadrilateral with $AC \perp BD$; $AC$ meets $BD$ at $E$. Prove that \[ EA^2 + EB^2 + EC^2 + ED^2 = 4 R^2 \] where $R$ is the radius of the circumscribing circle.

At least one of the coefficients of a polynomial $P(x)$ is negative. Can all of the coefficients of all of its powers $(P(x))^n$, $n > 1$, be positive? (0 Kryzhanovskij)

Let $\Omega$ be a circle with center $O$. Let $\omega_1$ and $\omega_2$ be circles with centers $O_1$ and $O_2$, respectively, internally tangent to $\Omega$ at points $A$ and $B$, respectively, such that $O_1$ is on $\overline{OA}$, and $O_2$ is on $\overline{OB}$ and $\omega_1$. There exists a point $P$ on line $AB$ such that $P$ is on both $\omega_1$ and $\omega_2$. Let the external tangent of $\omega_1$ and $\omega_2$ on the same side of line $AB$ as $O$ hit $\omega_1$ at $X$ and $\omega_2$ at $Y$, and let lines $AX$ and $BY$ intersect at $N$. Given that $O_1X = 81$ and $O_2Y = 18$, the value of $NX \cdot NA$ can be written as $a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find $a+b+c$. [i]Proposed by Kevin Zhao[/i]

In trapezoid $ABCD$, $BC \parallel AD$, $AB = 13$, $BC = 15$, $CD = 14$, and $DA = 30$. Find the area of $ABCD$.

Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$. Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$, and the tangent at $B$ meet $AC$ at point $S_2$. Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$, $AB$ and $H_1H_2$ concur.

Source: 2018 Canadian Open Math Challenge Part C Problem 1 ----- At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center] $\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$, [color=transparent](A.)[/color]layer of this pyramid? $\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers. [color=transparent](B.)[/color]How many cans are on the bottom layer of the prism? $\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle. [color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...) [color=transparent](C.)[/color]For example, a prism could be composed of the following layers: [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center] Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.

For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$

a) Calculate the product $$\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)$$ b) Let the set $$A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}$$ Determine the sum of all products of $2$, of $4$, of $6$,... , of $2004$ ¸and of $ 2006$ different elements of the set $A$.

Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

Let $ ABC$ be a given triangle with the incenter $ I$, and denote by $ X$, $ Y$, $ Z$ the intersections of the lines $ AI$, $ BI$, $ CI$ with the sides $ BC$, $ CA$, and $ AB$, respectively. Consider $ \mathcal{K}_{a}$ the circle tangent simultanously to the sidelines $ AB$, $ AC$, and internally to the circumcircle $ \mathcal{C}(O)$ of $ ABC$, and let $ A^{\prime}$ be the tangency point of $ \mathcal{K}_{a}$ with $ \mathcal{C}$. Similarly, define $ B^{\prime}$, and $ C^{\prime}$. Prove that the circumcircles of triangles $ AXA^{\prime}$, $ BYB^{\prime}$, and $ CZC^{\prime}$ all pass through two distinct points.

A mother and her son are playing. At first, the son divides a ${300}\mathrm{\;g}$ wheel of cheese into 4 slices. Then the mother divides ${280}\mathrm{\;g}$ of butter between two plates. At last, the son puts the cheese slices on those plates. The son wins if on each plate the amount of cheese is not less than the amount of butter (otherwise the mother wins). Who of them can win irrespective of the opponent's actions? Alexandr Shapovalov

Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect? $ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Inside $\vartriangle ABC$, there is fixed a point $D$ such that $AC - DA > 1$ and $BC - BD > 1$. Prove that $EC - ED > 1$ for any point $E$ on segment $AB$.

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. [i]Proposed by Aryan Tajmir[/i]

Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is $19$.