Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$. [i]Proposed by Josef Tkadlec (Czech Republic)[/i]

Let $A=1,B=2,\ldots,Z=26$. Compute $BONANZA$, where the result is the product of the numbers represented by each letter. [i]2018 CCA Math Bonanza Lightning Round #1.1[/i]

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

A positive integer \(n\) is called perfect if the sum of its positive divisors \(\sigma(n)\) is twice \(n\), that is, \(\sigma(n) = 2n\). For example, \(6\) is a perfect number since the sum of its positive divisors is \(1 + 2 + 3 + 6 = 12\), which is twice \(6\). Prove that if \(n\) is a positive perfect integer, then: \[ \sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1} \] where the sums are taken over all prime divisors \(p\) of \(n\).

Find all ordered 6-tuples satisfy following system of modular equation: $ab+a'b' \equiv 1 $(mod 15) $bc+b'c' \equiv 1 $(mod 15) $ca+c'a' \equiv 1 $(mod 15) Given that $a,b,c,a',b',c' \epsilon (0;1;2;...;14)$

In triangle $ABC$, let $O$ be the circumcenter. The incircle of $ABC$ is tangent to $\overline{BC}$, $\overline{CA},$ and $\overline{AB}$ at points $D, E$, and $F$, respectively. Let $G$ be the centroid of triangle $DEF$. Suppose the inradius and circumradius of $ABC$ is $3$ and $8$, respectively. Over all such triangles $ABC$, pick one that maximizes the area of triangle $AGO$. If we write $AG^2 =\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then find $m$.

Let $ \sigma(\cdot): \mathbb{N}_0 \to \mathbb{N}_0$ be the function from every positive integer $ n$ to the sum of divisors $ \sum_{d \mid n}{d}$ (i.e. $ \sigma(6) \equal{} 6 \plus{} 3 \plus{} 2 \plus{} 1$ and $ \sigma(8) \equal{} 8 \plus{} 4 \plus{} 2 \plus{} 1$). Find all primes $ p$ such that $ p \mid \sigma(p \minus{} 1)$. [i](Salvatore Tringali)[/i]

Let n be a non-negative integer. Define the [i]decimal digit product[/i] \(D(n)\) inductively as follows: - If \(n\) has a single decimal digit, then let \(D(n) = n\). - Otherwise let \(D(n) = D(m)\), where \(m\) is the product of the decimal digits of \(n\). Let \(P_k(1)\) be the probability that \(D(i) = 1\) where \(i\) is chosen uniformly randomly from the set of integers between 1 and \(k\) (inclusive) whose decimal digit products are not 0. Compute \(\displaystyle\lim_{k\to\infty} P_k(1)\). [i]proposed by the ICMC Problem Committee[/i]

A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? $ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$. Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is [i]not[/i] marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane. (i) Can Evan construct* the reflection of $A$ over $\ell$? (ii) Can Evan construct the foot of the altitude from $A$ to $\ell$? *To construct a point, Evan must have an algorithm which marks the point in finitely many steps. [i]Proposed by Zack Chroman[/i]

Let be an $ n\times n $ positive-definite symmetric real matrix $ A. $ Prove the following equality. $$ \tiny\int_{\mathbb{R}^n} \exp\left( -\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}^\intercal A\begin{pmatrix} x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\right) dx_1dx_2\cdots dx_n=\normalsize\frac{\pi^{n/2}}{\sqrt{\det A} } $$

In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? [asy] import graph; size(9cm); pen dps = fontsize(10); defaultpen(dps); pair D = (0,0); pair F = (1/2,0); pair C = (1,0); pair G = (0,1); pair E = (1,1); pair A = (0,2); pair B = (1,2); pair H = (1/2,1); // do not look pair X = (1/3,2/3); pair Y = (2/3,2/3); draw(A--B--C--D--cycle); draw(G--E); draw(A--F--B); draw(D--H--C); filldraw(H--X--F--Y--cycle,grey); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$E$",E,E); label("$F$",F,S); label("$G$",G,W); label("$H$",H,N); label("$\displaystyle\frac12$",(0.25,0),S); label("$\displaystyle\frac12$",(0.75,0),S); label("$1$",(1,0.5),E); label("$1$",(1,1.5),E); [/asy] $ \textbf{(A)}\ \dfrac1{12}\qquad\textbf{(B)}\ \dfrac{\sqrt3}{18}\qquad\textbf{(C)}\ \dfrac{\sqrt2}{12}\qquad\textbf{(D)}\ \dfrac{\sqrt3}{12}\qquad\textbf{(E)}\ \dfrac16 $

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence? $ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $

A rope of length 10 [i]m[/i] is tied tautly from the top of a flagpole to the ground 6 [i]m[/i] away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 [i]cm/min[/i]. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)

Let $ABC$ be a triangle and $\Gamma$ the $A$- exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\Gamma$ with the lines $AB$ and $AC$, respectively. Let $S$ be the area of the quadrilateral $ADJE$, Find the maximum value that $\frac{S}{AJ^2}$ has and when equality holds.

Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Lewis Chen[/i]

Can a set of six numbers $\left\{a, b,c, \frac{a^2}{b} , \frac{b^2}{c} , \frac{c^2}{a} \right\}$ , where $a, b, c$ positive numbers, turn out to be exactly exactly three different numbers?

A mathematics teacher writes a quadratic equation on the blackboard of the form $$x^2+mx \star n = 0$$, with $m$ and $n$ integers. The sign of $n$ is blurred. Even so, Claudia solves it and obtains integer solutions, one of which is $2011$. Find all possible values of $m$ and $n$.

Let $P$ be a set of $n \ge 2$ distinct points in the plane, which does not contain any triplet of aligned points. Let $S$ be the set of all segments whose endpoints are points of $P$. Given two segments $s_1, s_2 \in S$, we write $s_1 \otimes s_2$ if the intersection of $s_1$ with $s_2$ is a point other than the endpoints of $s_1$ and $s_2$. Prove that there exists a segment $s_0 \in S$ such that the set $\{s \in S | s_0 \otimes s\}$ has at least $\frac{1}{15}\dbinom{n-2}{2}$ elements

Can we build parallelepiped $6 \times 7 \times 7$ from $1 \times 1 \times 2$ bricks, such that we have same amount bricks of one of $3$ directions ?

For each natural $n \ge 4$, find the smallest natural number $k$ that satisfies following condition: For an arbitrary arrangement of $k$ chips of two colors on $n\times n$ board, there exists a non-empty set such that all columns and rows contain even number ($0$ is also possible) of chips each color.

Find all infinite arithmetic progressions formed with positive integers such that there exists a number $N \in \mathbb{N}$, such that for any prime $p$, $p > N$, the $p$-th term of the progression is also prime.

Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.

Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

$6$ indistinguishable coins are given, $4$ are authentic, all of the same weight, and $2$ are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights. Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.