Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as: $u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$ a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$

Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$. [list=a] [*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$. [*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$. [/list]

Prove that for any positive integer $ k$, there exists an arithmetic sequence $ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3}, ... ,\frac{a_k}{b_k}$ of rational numbers, where $ a_i, b_i$ are relatively prime positive integers for each $ i \equal{} 1,2,...,k$ such that the positive integers $ a_1, b_1, a_2, b_2, ..., a_k, b_k$ are all distinct.

Evin picks distinct points $A, B, C, D, E$, and $F$ on a circle. What is the probability that there are exactly two intersections among the line segments $AB$, $CD$, and $EF$? [i]Proposed by Evin Liang[/i]

The plane is divided into unit cells, and each of the cells is painted in one of two given colors. Find the minimum possible number of cells in a figure consisting of entire cells which contains each of the $16$ possible colored $2\times2$ squares.

A tennis league has three teams, and each team plays the each of the other two teams twice. How many total matches are there, between these three tennis teams? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }12$

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

In triangle $ABC$, $AB=13$, $BC=15$, and $CA=14$. Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac pq$ where $p$ and $q$ are relatively prime positive integers, find $q.$

Find the number of eight-digit numbers the sum of whose digits is $4$

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

$3m$ balls numbered $1, 1, 1, 2, 2, 2, 3, 3, 3, \ldots, m, m, m$ are distributed into $8$ boxes so that any two boxes contain identical balls. Find the minimal possible value of $m$.

The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ? $ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$

A running track is the ring formed by two concentric circles. It is $ 10$ feet wide. The circumference of the two circles differ by about: $ \textbf{(A)}\ 10\text{ feet} \qquad\textbf{(B)}\ 30\text{ feet} \qquad\textbf{(C)}\ 60\text{ feet} \qquad\textbf{(D)}\ 100\text{ feet} \\ \textbf{(E)}\ \text{none of these}$

Two ships sailed at constant speeds on constant courses at see (its surface is considered to be flat). Their mutual distance was 20 nautical miles at 9:00 a.m., 15 miles at 9:35 a.m. and 13 miles at 9:55 a.m. a) Determine the square of their distance as a function of time. b) Find out when the ships were closest to each other and what was the distance.

$n$ - some natural. We write on the board all such numbers $d$, that $d\leq 1000$ and $d|n+k$ for some $ 1\leq k \leq 1000$. Let $S(n)$ -sum of all written numbers. Prove , that $S(n)<10^6$ and $S(n)>10^6$ has infinitely many solutions.

Let $N$ be the sum of all binomial coefficients $\binom{a}{b}$ such that $a$ and $b$ are nonnegative integers and $a+b$ is an even integer less than 100. Find the remainder when $N$ is divided by 144. (Note: $\binom{a}{b} = 0$ if $a<b$, and $\binom{0}{0} = 1$.)

Let the complex function $F(z)$ be regular on the punctuated disk $\{ 0<|z| < R\}$. By a [i]level curve[/i] we mean a component of the level set of $\mathrm{Re}F(z)$, that is, a maximal connected set on which $\mathrm{Re}F(z)$ is constant. Denote by $A(r)$ the union of those level curves that are entirely contained in the punctuated disk $\{ 0<|z|<r\}$. Prove that if the number of components of $A(r)$ has an upper bound independent of $r$ then $F(z)$ can only have a pole type singularity at $0$.

For a $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1 $ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $ AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.

Find a pair $\left(x,y\right)$ of positive integers $x<y$ such that $$37^2+46^2+49^2-20^2-17^2=x^2+y^2.$$ Note: there may be several answers; just provide one of them. [i]2017 CCA Math Bonanza Individual Round #14

Find all triples $(a,b,c)$ of positive integers with the following property: for every prime $p$, if $n$ is a quadratic residue $\mod p$, then $an^2+bn+c$ is a quadratic residue $\mod p$.

There is a ruble coin in each cell of the board with $2\times 200$. Dasha and Sonya play, taking turns making moves, Dasha starts. In one move, it is allowed to select any coin and move it: Dasha moves the coin to a diagonally adjacent cell, Sonya is to the side adjacent. If two coins end up in the same cell, one of them is immediately removed from the board and goes to Sonya. Sonya can stop the game at any time and take all the coins she has received. What is the biggest win she can get, no matter how she plays Dasha? [i]A. Kuznetsov[/i]

[u]Round 5[/u] [b]p13.[/b] Given that $x^3 + y^3 = 208$ and $x + y = 4$, what is the value of $\frac{1}{x} +\frac{1}{y}$? [b]p14.[/b] Find the sum of all three-digit integers $n$ such that the value of $n$ is equal to the sum of the factorials of $n$’s digits. [b]p15.[/b] Three christmas lights are initially off. The Grinch decides to fiddle around with the lights, switching one of the lights each second. He wishes to get every possible combination of lights. After how many seconds can the Grinch complete his task? [u]Round 6[/u] [b]p16.[/b] A regular tetrahedron of side length $1$ has four similar tetrahedrons of side length $1/2$ chopped off, one from each of the four vertices. What is the sum of the numbers of vertices, edges, and faces of the remaining solid? [b]p17.[/b] Mario serves a pie in the shape of a regular $2013$-gon. To make each slice, he must cut in a straight line starting from one vertex and ending at another vertex of the pie. Every vertex of a slice must be a vertex of the original $2013$-gon. If every person eats at least one slice of pie regardless of the size, what is the maximum number of people the $2013$-gon pie can feed? [b]p18.[/b] Find the largest integer $x$ such that $x^2 + 1$ divides $x^3 + x - 1000$. [u]Round 7[/u] [b]p19.[/b] In $\vartriangle ABC$, $\angle B = 87^o$, $\angle C = 29^o$, and $AC = 37$. The perpendicular bisector of $\overline{BC}$ meets $\overline{AC}$ at point $T$. What is the value of $AB + BT$? [b]p20.[/b] Consider the sequence $f(1) = 1$, $f(2) = \frac12$ ,$ f(3) =\frac{1+3}{2}$, $f(4) =\frac{ 1+3}{2+4}$ ,$ f(5) = \frac{ 1+3+5}{2+4} . . . $ What is the minimum value of $n$, with $n > 1$, such that $|f(n) - 1| \le \frac{1}{10 }$. [b]p21.[/b] Three unit circles are centered at $(0, 0)$,$(0, 2)$, and $(2, 0)$. A line is drawn passing through $(0, 1)$ such that the region inside the circles and above the line has the same area as the region inside the circles and below the line. What is the equation of this line in $y = mx + b$ form? [u]Round 8[/u] [b]p22.[/b] The two walls of a pinball machine are positioned at a $45$ degree angle to each other. A pinball, represented by a point, is fired at a wall (but not at the intersection of the two walls). What is the maximum number of times the ball can bounce off the walls? [b]p23.[/b] Albert is fooling people with his weighted coin at a carnival. He asks his guests to guess how many times heads will show up if he flips the coin $4$ times. Richard decides to play the game and guesses that heads will show up $2$ times. In the previous game, Zach guessed that the heads would show up 3 times. In Zach’s game, there were least 3 heads, and given this information, Zach had a $\frac49$ chance of winning. What is the probability that Richard guessescorrectly? [b]p24.[/b] Let $S$ be the set of all positive integers relatively prime to $2013$ that have no prime factor greater than $15$. Find the sum of the reciprocals of all of the elements of $S$. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134546p28406927]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3137069p28442224]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Peter and Basil play the following game on a horizontal table $1\times{2019}$. Initially Peter chooses $n$ positive integers and writes them on a board. After that Basil puts a coin in one of the cells. Then at each move, Peter announces a number s among the numbers written on the board, and Basil needs to shift the coin by $s$ cells, if it is possible: either to the left, or to the right, by his decision. In case it is not possible to shift the coin by $s$ cells neither to the left, nor to the right, the coin stays in the current cell. Find the least $n$ such that Peter can play so that the coin will visit all the cells, regardless of the way Basil plays.

Find the probability such that when a polynomial in $\mathbb Z_{2027}[x]$ having degree at most $2026$ is chosen uniformly at random, $$x^{2027}-x | P^{k}(x) - x \iff 2021 | k $$ (note that $2027$ is prime). Here $P^k(x)$ denotes $P$ composed with itself $k$ times. [i]Proposed by Grant Yu[/i]

Find the number of functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ for which $f(h+k)+f(hk)=f(h)f(k)+1$, for all integers $h$ and $k$.