Prove that every convex $n$-gon of area $1$ contains a quadrilateral of area at least $\frac12 $. .

You are trying to maximize a function of the form $f(x, y, z) = ax + by + cz$, where $a$, $b$, and $c$ are constants. You know that $f(3, 1, 1) > f(2, 1, 1)$, $f(2, 2, 3) > f(2, 3, 4)$, and $f(3, 3, 4) > f(3, 3, 3)$. For $-5 \le x,y,z \le 5$, what value of $(x,y,z)$ maximizes the value of $f(x, y, z)$? Give your answer as an ordered triple.

Let $k>2$ be a positive integer. Elise and Xavier play a game that has four steps, in this order. [list=1] [*]Elise picks $2$ nonzero digits $(1-9)$, called $e$ and $f$. [*]Xavier then picks $k$ nonzero digits $(1-9)$, called $x_1,\cdots,x_k$. [*]Elise picks any positive integer $d$. [*]Xaiver picks an integer $b>10$.[/list] Each player's choices are known to the other player when the choices are made. The winner is determined as follows. Elise writes down the two-digit base $b$ number $ef_b$. Next, Xavier writes the $k$-digit base $b$ number that is constructed by concatenating his digits, \[(x_1\cdots x_k)_b.\] They then compute the greatest common divisor (gcd) of these two numbers. If this gcd is greater than or equal to the integer $d$ then Xavier wins. Otherwise Elise wins. (As an example game for $k=3$, Elise chooses the digits $(e, f) = (2, 4)$, Xavier chooses $(4, 4, 8)$, and then Elise picks $d = 100$. Xavier picks base $b = 25$. The base-25 numbers $2425$ and $44825$ are, respectively, equal to $54$ and $2608$. The greatest common divisor of these two is $2$, which is much less than $100$, so Elise wins handily.) Find all $k$ for which Xavier can force a win, no matter how Elise plays.

An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

In the [i]minesweeper[/i] game below, each unopened square (for example, the one in the top left corner) is either empty or contains a mine. The other squares are empty and display the number of mines in the neighboring 8 squares (if this is 0, the square is unmarked). What is the minimum possible number of mines present on the field? [img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9hLzNlYTNhMjI2YWYyNmFkZGFiNWFmODBhNzA3YjA3OWM5MTZlNDlkLnBuZw==&rn=bWluZXN3ZWVwZXIucG5n[/img] [i]2016 CCA Math Bonanza Team #4[/i]

Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\cdots +a_{2004}$, where $1\le a_1<a_2<\cdots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$. [i]Chen Yonggao[/i]

The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is: $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad \textbf{(E) }10$

If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by $x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively, when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals $\textbf{(A) }\frac{1}{256}\qquad\textbf{(B) }-\frac{1}{16}\qquad\textbf{(C) }1\qquad\textbf{(D) }-16\qquad\textbf{(E) }256$

If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}} $$

In a triangle $ABC$, the line symmetric to the median through $A$ with respect to the bisector of the angle at $A$ intersects $BC$ at $M$. Points $P$ on $AB$ and $Q$ on $AC$ are chosen such that $MP\parallel AC$ and $MQ\parallel AB$. Prove that the circumcircle of the triangle $MPQ$ is tangent to the line $BC$.

Find the sum of the smallest 20 positive real solutions of the equation $\sin x=\cos 2x .$

Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$

In a triangle $ ABC $ that has perimeter $ P, $ prove that it's isosceles if and only if $$ P^2+\sin^2 (\angle ABC-\angle BCA) =4\cdot AB\cdot AC\cdot\cos^2\frac{\angle ABC}{2}\cdot\cos^2\frac{\angle BCA}{2} . $$

Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point. [i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]

A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide. The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?

A robot on the number line starts at $1$. During the first minute, the robot writes down the number $1$. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by $n/t$, where $n$ is the number it just moved to, and $t$ is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down $2/2=1$. Find the expected sum of all numbers it writes down, given that it is finite. [i]Proposed by Ethan Tan[/i]

If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.

Jenn is competing in a puzzle hunt with six regular puzzles and one additional meta-puzzle. Jenn can solve any puzzle regularly. Additionally, if she has already solved the meta-puzzle, Jenn can also back-solve a puzzle. A back-solve is distinguishable from a regular solve. The meta puzzle cannot be the first puzzle solved. How many possible solve orders for the seven puzzles are possible? For example, Jenn may solve #3, solve #5, solve #6, solve the meta-puzzle, solve #2, solve #1, and then solve #4. However, she may not solve #2, solve #4, solve #6, back-solve #1, solve #3, solve #5, and then solve the meta-puzzle.

A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does $2022$ of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?

Given a sequence of numbers $a_1, a_2, ..., a_{15}$, one can always construct a new sequence $b_1,b_2, ..., b_{15}$, where $b_i$ is equal to the number of terms in the sequence $\{a_k\}^{15}_{k=1}$ less than $a_i$ ($i = 1, 2,..., 15$). Is there a sequence $\{a_k\}^{15}_{k=1}$ for which the sequence $\{b_k\}^{15}_{k=1}$ is $$1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13 \,?$$

Let $S$ be the set of positive integers $n$ such that the inequality \[ \phi(n) \cdot \tau(n) \geq \sqrt{\frac{n^3}{3}} \] holds, where $\phi(n)$ is the number of positive integers $k \le n$ that are relatively prime to $n$, and $\tau(n)$ is the number of positive divisors of $n$. Prove that $S$ is finite.

For what values of the ratio $a/b$ is the limaçon $r = a - b \cos \theta$ a convex curve? $(a > b > 0)$

A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is $ \textbf{(A)}\ \$306 \qquad \textbf{(B)}\ \$333 \qquad \textbf{(C)}\ \$342 \qquad \textbf{(D)}\ \$348 \qquad \textbf{(E)}\ \$360$

$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x \plus{} y^2)(x^2 \minus{} y)/(xy)$ ? $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17$

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]