Let $ABC$ be an acute scalene triangle. The bisector of the angle $\angle ABC$ intersects the altitude $AD$ at $K$. Let $M$ be the projection of $B$ onto $CK$ and let $N$ be the intersection between $BM$ and $AK$. Let $T$ be a point on $AC$ such that $NT$ is parallel to $DM$. Prove that $BM$ is the bisector of the angle $\angle TBC$. [i]Melih Üçer, Turkey[/i]

An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies \[ x_{n+2}=\gcd(x_{n+1},x_n)+2006 \] for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?

How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? $\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$

How many positive integers $x$ less than $10 000$ are there such that $2^x - x^2$ is divisible by $7$ ?

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

Prove that there exists an undirected graph having $ 2004 $ vertices such that for any $ \in\{ 1,2,\ldots ,1002 \} , $ there exists at least two vertices whose orders are $ n. $

Recall that an arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is the same. Suppose $x_1$, $x_2$, $x_3$ forms an arithmetic sequence. If $x_2 = 2023$, compute $x_1 + x_2 + x_3$.

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and centroid $G$. Let $X$ be the intersection of the line tangent to the circumcircle of $\triangle ABC$ at $A$ and the line perpendicular to $GO$ at $G$. Let $Y$ be the intersection of lines $XG$ and $BC$. Given that the measures of $\angle ABC, \angle BCA, $ and $\angle XOY$ are in the ratio $13 : 2 : 17, $ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] unitsize(5mm); pair A,B,C,X,G,O,Y; A = (2,8); B = (0,0); C = (15,0); dot(A,5+black); dot(B,5+black); dot(C,5+black); draw(A--B--C--A,linewidth(1.3)); draw(circumcircle(A,B,C)); O = circumcenter(A,B,C); G = (A+B+C)/3; dot(O,5+black); dot(G,5+black); pair D = bisectorpoint(O,2*A-O); pair E = bisectorpoint(O,2*G-O); draw(A+(A-D)*6--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10)); draw(intersectionpoint(G--G+(G-E)*10,B--C)--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10)); X = intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10); Y = intersectionpoint(G--G+(G-E)*10,B--C); dot(Y,5+black); dot(X,5+black); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,SE); label("$O$",O,ESE); label("$G$",G,W); label("$X$",X,dir(0)); label("$Y$",Y,NW); draw(O--G--O--X--O--Y); markscalefactor = 0.07; draw(rightanglemark(X,G,O)); [/asy]

Let $a, b, c$ be non-negative reals which satisfy $a+b+c=1$. Prove that $\frac{\sqrt{a}}{b+1}+\frac{\sqrt{b}}{c+1}+\frac{\sqrt{c}}{a+1}>\frac{1}{2}(\sqrt{a}+\sqrt{b}+\sqrt{c})$

Let $A_1A_2A_3A_4A_5$ be a convex, cyclic pentagon with $\angle A_i + \angle A_{i+1} >180^{\circ}$ for all $i \in \{1,2,3,4,5\}$ (all indices modulo $5$ in the problem). Define $B_i$ as the intersection of lines $A_{i-1}A_i$ and $A_{i+1}A_{i+2}$, forming a star. The circumcircles of triangles $A_{i-1}B_{i-1}A_i$ and $A_iB_iA_{i+1}$ meet again at $C_i \neq A_i$, and the circumcircles of triangles $B_{i-1}A_iB_i$ and $B_iA_{i+1}B_{i+1}$ meet again at $D_i \neq B_i$. Prove that the ten lines $A_iC_i, B_iD_i$, $i \in \{1,2,3,4,5\}$, have a common point.

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$? [asy] import cse5;pathpen=black;pointpen=black; size(2inch); pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162); D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5)); D(A--C--E--B--D--cycle); pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A); D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle); [/asy] $\textbf{(A) } 3 \qquad\textbf{(B) } 12-4\sqrt5 \qquad\textbf{(C) } \dfrac{5+2\sqrt5}{3} \qquad\textbf{(D) } 1+\sqrt5 \qquad\textbf{(E) } \dfrac{11+11\sqrt5}{10} $

Maggie Waggie organizes a pile of 127 calculus tests in alphabetical order, with Joccy Woccy's test being 64th in the pile. While Maggie isn't looking, Joccy walks over and randomly scrambles the entire pile of tests. When Maggie returns, she is oblivious to the fact that Joccy has tampered with the list. She uses a binary search algorithm to find Joccy's test, where she looks at the test in the middle of the pile. If the test is not Joccy's, she binary searches the top half of the list if the test appears after Joccy's name when arranged alphabetically, or the bottom half of the list otherwise. The probability that Maggie finds Joccy's test can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #5[/i]

A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.

Find the smallest positive integer $n$ for which $315^2-n^2$ evenly divides $315^3-n^3$. [i]Proposed by Kyle Lee[/i]

All positive differences $a_i -a_j$ of five different positive integers $a_1$, $a_2$, $a_3$, $a_4$ and $a_5$ are all different. Let $A$ be the set formed by the largest elements of each group of $5$ elements that meet said condition. Determine the minimum element of $A$.

Show that every number of the form $2^{p}3^{q}$ , where $p,q$ are nonnegative integers, divides some number of the form $a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}$, where $a_{2i}\in\{1,2,...,9\}$

Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number n such that $\sigma_1(n) = 1854$?

Compute the number of ways to tile a $3\times5$ rectangle with one $1\times1$ tile, one $1\times2$ tile, one $1\times3$ tile, one $1\times4$, and one $1\times5$ tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.)

Let be $\gamma_1$ a circumference of radius $r$ and $P$ an exterior point that is at distance $a$ from the centre of $\gamma_1$. We build two tangent lines $r,s$ to $\gamma_1$ from $P$ and $\gamma_2$ is constructed as a smaller circumference, tangent to both lines and, also, tangent to $\gamma_1$. We construct inductively $\gamma_{n+1}$ as a tangent circumference to $\gamma_{n}$ and, also, tangent to $r$ and $s$. Determine: a) The radius of $\gamma_2$. b) The radius of $\gamma_n$. c) The sum of the lengths of $\gamma_1, \gamma_2, \gamma_3, ...$.

A polygon of area $S$ is contained inside a square of side length $a$. Show that there are two points of the polygon that are a distance of at least $S/a$ apart.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

Let $n,d$ be integers with $n,k > 1$ such that $g.c.d(n,d!) = 1$. Prove that $n$ and $n+d$ are primes if and only if $$d!d((n-1)!+1) + n(d!-1) \equiv 0 \hspace{0.2cm} (\bmod n(n+d)).$$

In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?

A game in a casino uses the diagram shown. At the start a ball appears at $S$. Each time the player presses a button, the ball moves to one of the adjacent letters with equal probability. The game ends when one of the following two things happens: (i) The ball returns to $S$, the player loses. (ii) The ball reaches $G$, the player wins. Find the probability that the player wins and the expected duration of a game.

What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?