Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$

Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16$

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$, $BCP$ and $OMN$, respectively. The intersection point of $\omega_1$ and $\omega_3$, which is not on the arc $APD$ of $\omega_1$, is $E$ and the intersection point of $\omega_2$ and $\omega_3$, which is not on the arc $BPC$ of $\omega_2$, is $F$. Prove that $OF=OE$. Proposed by Seyed Amirparsa Hosseini Nayeri

The sequence $ \{x_n\}$ is defined by \[ \left\{ \begin{array}{l}x_1 \equal{} \frac{1}{2} \\x_n \equal{} \frac{{\sqrt {x_{n \minus{} 1} ^2 \plus{} 4x_{n \minus{} 1} } \plus{} x_{n \minus{} 1} }}{2} \\\end{array} \right.\] Prove that the sequence $ \{y_n\}$, where $ y_n\equal{}\sum_{i\equal{}1}^{n}\frac{1}{{{x}_{i}}^{2}}$, has a finite limit and find that limit.

Integers $1,2, ...,2n$ are arbitrarily assigned to boxes labeled with numbers $1, 2,..., 2n$. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo $2n$.

[b]8.[/b] Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? ([b]G.36[/b]) [L. Fejes-Tóth]

Let $ABC$ be a triangle with $AB=20$ and $AC=18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE=AF=8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC=m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $100m+n$. [i] Proposed by James Lin[/i]

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee[/i]

In acute triangle $ABC,$ the feet of the altitudes are $A_1,B_1,$ and $C_1$ (with the usual notations on sides $BC,CA,$ and $AB$ respectively). The circumcircles of triangles $AB_1C_1$ and $BC_1A_1$ intersect at the circumcircle of triangle $ABC$ ar points $P\neq A$ and $Q\neq B,$ respectively. Prove that lines $AQ, BP$ and the Euler line of triangle $ABC$ are either concurrent or parallel to each other. [i]Proposed by Géza Kós, Budapest[/i]

Consider the rectangle in the coordinate plane with corners $(0, 0)$, $(16, 0)$, $(16, 4)$, and $(0, 4)$. For a constant $x_0 \in [0, 16]$, the curves \[ \{(x, y) : y = \sqrt{x} \,\text{ and }\, 0 \leq x \leq 16\} \quad \text{and} \quad \{(x_0, y) : 0 \leq y \leq 4\} \] partition this rectangle into four 2D regions. Over all choices of $x_0$, determine the smallest possible sum of the areas of the bottom-left and top-right 2D regions in this partition. (The bottom-left region is $\{(x, y) : 0 \leq x < x_0 \,\text{ and }\, 0 \leq y < \sqrt{x}\}$, and the top-right region is $\{(x, y) : x_0 < x \leq 16 \,\text{ and }\, \sqrt{x} < y \leq 4\}$.)

Consider the circles $\omega$, $\omega_1$, $\omega_2$, where $\omega_1$, $\omega_2$ pass through the center $O$ of $\omega$. The circle $\omega$ cuts $\omega_1$ at $A, E$ and $\omega_2$ at $C, D$. The circles $\omega_1$ and $\omega_2$ intersect at $O$ and $M$. If A$D$ cuts $CE$ at $B$ and if $MN \perp BO$, ($N \in BO$) prove that $2MN^2 \le BM \cdot MO$.

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

What is the smallest positive integer $n$ such that there exists a choice of signs for which \[1^2\pm2^2\pm3^2\ldots\pm n^2=0\] is true? [i]2019 CCA Math Bonanza Team Round #5[/i]

5. Given the point T in rectangle ABCD, the distances from T to A,B,C is 15,20,25. Find the distance from T to D.

Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that (a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence); (b) if the sequence is periodic, then the first number is rational. (G Shabat)

At the two ends $A, B$ of a diameter (of length $2r$) of a pavement horizontal circular rise two vertical columns, of equal height h, whose ends support a beam $A' B' $ of length equal to the before mentioned diameter. It forms a covered by placing numerous taut cables (which are admitted to be rectilinear), joining points of the beam $A'B'$ with points of the circumference edge of the pavement, so that the cables are perpendicular to the beam $A'B'$ . You want to find out the volume enclosed between the roof and the pavement. [hide=original wording]En los dos extremos A, B de un di´ametro (de longitud 2r) de un pavimento circular horizontal se levantan sendas columnas verticales, de igual altura h, cuyos extremos soportan una viga A' B' de longitud igual al diametro citado. Se forma una cubierta colocando numerosos cables tensos (que se admite que quedan rectilıneos), uniendo puntos de la viga A'B' con puntos de la circunferencia borde del pavimento, de manera que los cables queden perpendiculares a la viga A'B' . Se desea averiguar el volumen encerrado entre la cubierta y el pavimento.[/hide]

Find all functions $f : \mathbb{R} \mapsto \mathbb{R} $ such that $$ \left(f(x)+y\right)\left(f(x-y)+1\right)=f\left(f(xf(x+1))-yf(y-1)\right)$$ for all $x,y \in \mathbb{R}$

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes.

An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. The aquarium is filled with water to a depth of $ 37$ cm. A rock with volume $ 1000 \text{cm}^3$ is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise? $ \textbf{(A)}\ 0.25 \qquad \textbf{(B)}\ 0.5 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 1.25 \qquad \textbf{(E)}\ 2.5$

[i]Triangular numbers[/i] are numbers of the form $1 + 2 + . . . + n$ with positive integer $n$, that is $1, 3, 6, 10$, . . . . Find the largest non-triangular positive integer number that cannot be represented as the sum of distinct triangular numbers. [i]Proposed by A. Golovanov[/i]

Calculate the limit of the following sequences: [b]a)[/b] n^{n!}/(n!)^n [b]b)[/b] n^{ln n}/n! [i]Adrian Troie[/i]

In a scalene triangle $ABC$, $H$ is the orthocenter, and $G$ is the centroid. Let $A_b$ and $A_c$ be points on $AB$ and $AC$, respectively, such that $B$, $C$, $A_b$, $A_c$ are cyclic, and the points $A_b$, $A_c$, $H$ are collinear. $O_a$ is the circumcenter of the triangle $AA_bA_c$. $O_b$ and $O_c$ are defined similarly. Prove that the centroid of the triangle $O_aO_bO_c$ lies on the line $HG$.