Isosceles triangle $\triangle{ABC}$ has $\angle{BAC}=\angle{ABC}=30^\circ$ and $AC=BC=2$. If the midpoints of $BC$ and $AC$ are $M$ and $N$, respectively, and the circumcircle of $\triangle{CMN}$ meets $AB$ at $D$ and $E$ with $D$ closer to $A$ than $E$ is, what is the area of $MNDE$? [i]2019 CCA Math Bonanza Individual Round #9[/i]

Let $ABCDA_1B_1C_1D_1$ be a cube. On the sides $AB{}$ and $AD{}$ there are the points $M{}$ and $N{}$, respectively, such that $AM+AN=AB$. Show that the measure of the dihedral angle between the planes $(MA_1C)$ and $(NA_1C)$ doe not depend on the positions of $M{}$ and $N{}$. Find this measure.

Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$. (A. Levin)

Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$. Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$, there exist at least $t$ elements of $A_i$, such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$. Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$.

$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$.

Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.

Find all irrational real numbers $\alpha$ such that \[ \alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha \] are both rational numbers.

$ABCD$ is a cyclic quadrilateral. Lines $AB,CD$ intersect at $E$, lines $AD,BC$ intersect at $F$, and $EM$ and $FN$ are tangents to the circumcircle of $ABCD$. Two circles are constructed with $E,F$ their centers and $EM, FN$ their radii, respectively. $K$ is one of their intersections. Prove that $EK$ is perpendicular to $FK$.

A sequence of positive integers $a_1, a_2, \ldots$ satisfies $a_k + a_l = a_m + a_n$ for all positive integers $k,l,m,n$ satisfying $kl = mn$. Prove that if $p$ divides $q$ then $a_p \le a_q$.

Let $n_1$, $n_2$, $\ldots$, $n_{1998}$ be positive integers such that \[ n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2. \] Show that at least two of the numbers are even.

The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy $a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}$ and $b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}$ for every integer $n = 1, 2, \dots, 2013$. If $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute \[ \sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right). \][i]Proposed by Evan Chen[/i]

Find all pairs of real numbers $(x, y)$ satisfying the following system of equations $2-x^{3}=y, 2-y^{3}=x$.

Function $f:\mathbb{R}\to\mathbb{R}$, satisfies that $f(0)=1$, and $f(xy+1)=f(x)f(y)-f(y)-x+2$, then $f(x)=$________.

Let $A_0$, $A_1$, $A_2$, ..., $A_n$ be nonnegative numbers such that \[ A_0 \le A_1 \le A_2 \le \dots \le A_n. \] Prove that \[ \left| \sum_{i = 0}^{\lfloor n/2 \rfloor} A_{2i} - \frac{1}{2} \sum_{i = 0}^n A_i \right| \le \frac{A_n}{2} \, . \] (Note: $\lfloor x \rfloor$ means the greatest integer that is less than or equal to $x$.)

Given the sequence $(u_n)$ satisfying:$$\left\{ \begin{array}{l} 1 \le {u_1} \le 3\\ {u_{n + 1}} = 4 - \dfrac{{2({u_n} + 1)}}{{{2^{{u_n}}}}},\forall n \in \mathbb{Z^+}. \end{array} \right.$$ Prove that: $1\le u_n\le 3,\forall n\in \mathbb{Z^+}$ and find the limit of $(u_n).$

A square is divided into $5$ rectangles in such a way that its $4$ vertices belong to $4$ of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square. [img]https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png[/img]

Given a prime $p$ and positive integer $k$, an integer $n$ with $0 \le n < p$ is called a $(p, k)$-Hofstadterian residue if there exists an infinite sequence of integers $n_0, n_1, n_2, \ldots$ such that $n_0 \equiv n$ and $n_{i + 1}^k \equiv n_i \pmod{p}$ for all integers $i \ge 0$. If $f(p, k)$ is the number of $(p, k)$-Hofstadterian residues, then compute $\displaystyle \sum_{k = 1}^{2016} f(2017, k)$. [i]Proposed by Ashwin Sah[/i]

A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.

Given a triangle $ABC$ with circumcircle $\Gamma$, and two arbitrary points $X, Y$ on $\Gamma$. Let $D$, $E$, $F$ be points on lines $BC$, $CA$, $AB$, respectively, such that $AD$, $BE$, and $CF$ concur at a point $P$. Let $U$ be a point on line $BC$ such that $X$, $Y$, $D$, $U$ are concyclic. Similarly, let $V$ be a point on line $CA$ such that $X$, $Y$, $E$, $V$ are concyclic, and let $W$ be a point on line $AB$ such that $X$, $Y$, $F$, $W$ are concyclic. Prove that $AU$, $BV$, $CW$ concur at a single point. [i]Proposed by chengbilly[/i]

Let $ABC$ be a non-equilateral triangle in the plane, and let $T$ be a point different from its vertices. Define $A_T$, $B_T$ and $C_T$ as the points where lines $AT$, $BT$, and $CT$ meet the circumcircle of $ABC$. Prove that there are exactly two points $P$ and $Q$ in the plane for which the triangles $A_PB_PC_P$ and $A_QB_QC_Q$ are equilateral. Prove furthermore that line $PQ$ contains the circumcenter of $\triangle ABC$.

The SET game is a deck with $81$ unique cards that vary in four features across three possibilities for each kind of feature: shape (oval, squiggle or diamond), color (red, green or purple), number of shapes (one, two or three) and shading (solid, striped or open). A $\textbf{set}$ consists in three cards whose characteristics, when considered individually, are the same on each card or different on all of them. All features have to satisfy this rule. In other words: the shape must be the same on all three cards or different on all them, the color must be the same on the three cards or different on all them, and so on. Ana and Bárbara divided among themselves the $81$ SET cards. Ana got $40$ cards and Bárbara got $41$. Each girl counted the number of ways she could form a three-card $\textbf{set}$ with her cards. What are the possible values of the sum of these two numbers?

Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x>y$ and $x$ is to the left of $y$, and replaces the pair $(x,y)$ by either $(y+1,x)$ or $(x-1,x)$. Prove that she can perform only finitely many such iterations. [i]Proposed by Warut Suksompong, Thailand[/i]

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

Solve the system of equations $$ \qquad<br /> \begin{array}{c}<br /> x_1x_2 = 1\\<br /> x_2x_3 = 2\\<br /> x_3x_4 = 3\\<br /> \ldots\\<br /> x_nx_1 = n<br /> \end{array}$$

Given are real numbers $a_1, a_2, \ldots, a_n$ ($n>3$), such that $a_k^3=a_{k+1}^2+a_{k+2}^2+a_{k+3}^2$ for all $k=1,2,...,n$. Prove that all numbers are equal.