Let $ a,\ b,\ c$ be postive constants. Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.

Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove that \[ \frac{1+ab}{1+a} + \frac{1+bc}{1+b} + \frac{1+cd}{1+c} + \frac{1+da}{1+d} \geq 4 . \] [i]Proposed by A. Khrabrov[/i]

Given an acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$. Let $K$ be a point inside $ABC$ which is not $O$ nor $H$. Point $L$ and $M$ are located outside the triangle $ABC$ such that $AKCL$ and $AKBM$ are parallelogram. At last, let $BL$ and $CM$ intersects at $N$, and let $J$ be the midpoint of $HK$. Show that $KONJ$ is also a parallelogram. [i]Raja Oktovin, Pekanbaru[/i]

Two families of parallel lines are given in the plane, consisting of $15$ and $11$ lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let $V$ be the set of $165$ intersection points of the lines under consideration. Show that there exist not fewer than $1986$ distinct squares with vertices in the set $V .$

Let $k$ and $n$ be positive integers and let $$S=\{(a_1,\ldots,a_k)\in \mathbb{Z}^{k}\;|\; 0\leq a_k\leq\cdots\leq a_1 \leq n,a_1+\cdots+a_k=n\}$$. Determine, with proof, the value of $$\sum_{(a_1,\ldots,a_k)\in S}\binom{n}{a_1}\binom{a_1}{a_2}\cdots\binom{a_{k-1}}{a_k}$$ in terms of $k$ and $n$, where the sum is over all $k$-tuples in $S$.

Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$. [i]Proposed by Jaroslaw Wroblewski, Poland[/i]

Solve in $\mathbb{R}$ the equation : $(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.

Suppose that $ m$ and $ n$ are odd integers such that $ m^2\minus{}n^2\plus{}1$ divides $ n^2\minus{}1$. Prove that $ m^2\minus{}n^2\plus{}1$ is a perfect square.

The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle $ABC$) $F$ is marked in triangle $ABC$. $A_1$ is on the side $BC$ such that $AA_1$ is the altitude of triangle $ABC$. Prove that the line symmetric to $FA_1$ with respect to $BC$ is perpendicular to $IO$, where $O$ is the center of the circumcircle of the triangle $ABC$ and $I$ is the center of its incircle.

Consider $2n+1$ points in space, no four of which are coplanar where $n>1$. Each line segment connecting any two of these points is either colored red, white or blue. A subset $M$ of these points is called a [i]connected monochromatic[/i] subset, if for each $a,b \in M$, there are points $a=x_0,x_1, \dots, x_l = b$ that belong to $M$ such that the line segments $x_0x_1, x_1x_2, \dots, x_{l-1}x_1$ are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic $k-$subset, find the largest value of $k$. ($l > 1$)

On an infinite chessboard (whose squares are labeled by $ (x, y)$, where $ x$ and $ y$ range over all integers), a king is placed at $ (0, 0)$. On each turn, it has probability of $ 0.1$ of moving to each of the four edge-neighboring squares, and a probability of $ 0.05$ of moving to each of the four diagonally-neighboring squares, and a probability of $ 0.4$ of not moving. After $ 2008$ turns, determine the probability that the king is on a square with both coordinates even. An exact answer is required.

Let $S=\{(a,b)|a=1,2,\dots,n,b=1,2,3\}$. A [i]rook tour[/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\dots,p_{3n}$ is sequence such that (i) $p_i\in S,$ (ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\le i<3n,$ (iii) for each $p\in S$ there is a unique $i$ such that $p_i=p.$ How many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$ (The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)

Find all pairs of integers $(x,y)$ such that $x \geq 0$ and \[ (6^x-y)^2 = 6^{x+1}-y. \]

In an equilateral trapezoid, the point $O$ is the midpoint of the base $AD$. A circle with a center at a point $O$ and a radius $BO$ is tangent to a straight line $AB$. Let the segment $AC$ intersect this circle at point $K(K \ne C)$, and let $M$ is a point such that $ABCM$ is a parallelogram. The circumscribed circle of a triangle $CMD$ intersects the segment $AC$ at a point $L(L\ne C)$. Prove that $AK=CL$.

Given that positive integers $a,b$ satisfy \[\frac{1}{a+\sqrt{b}}=\sum_{i=0}^\infty \frac{\sin^2\left(\frac{10^\circ}{3^i}\right)}{\cos\left(\frac{30^\circ}{3^i}\right)},\] where all angles are in degrees, compute $a+b$. [i]2021 CCA Math Bonanza Team Round #10[/i]

In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have.

Clara and Nick each randomly and independently pick an integer between $0$ and $2017$, inclusive. What is the probability that the two integers they pick sum to an even number?

A black pawn and a white pawn are placed on the first square and the last square of a $ 1\times n$ chessboard, respectively. Wiwit and Siti move alternatingly. Wiwit has the white pawn, and Siti has the black pawn. The white pawn moves first. In every move, the player moves her pawn one or two squares to the right or to the left, without passing the opponent's pawn. The player who cannot move anymore loses the game. Which player has the winning strategy? Explain the strategy.

In a tournament with $2015$ teams, each team plays every other team exactly once and no ties occur. Such a tournament is [i]imbalanced [/i]if for every group of $6$ teams, there exists either a team that wins against the other $5$ or a team that loses to the other $5$. If the teams are indistinguishable, what is the number of distinct imbalanced tournaments that can occur?

Suppose that positive integers $a_1,a_2,\ldots,a_{2014}$ (not necessarily distinct) satisfy the condition that: $\frac{a_1}{a_2},\frac{a_2}{a_3},\ldots,\frac{a_{2013}}{a_{2014}}$ are pairwise distinct. What is the minimal possible number of distinct numbers in $\{a_1,a_2,\ldots,a_{2014}\}$?

Michael and James are playing a game where they alternate throwing darts at a simplified dartboard. Each dart throw is worth either 25 points or 50 points. They track the sequence of scores per throw (which is shared between them), and on the first time the three most recent scores sum to 125, the person who threw the last dart wins. On each throw, a given player has a $2/3$ chance of getting the score they aim for, and a $1/3$ chance of getting the other score. Suppose Michael goes first, and the first two throws are both 25. If both players use an optimal strategy, what is the probability Michael wins? [i]Proposed by Michael Duncan[/i]

Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.

Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]