There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$? $\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Solve for $x$. $$\frac{1}{2-\frac{3}{4-x}}=5$$ $\text{(A) }-1\qquad\text{(B) }2\qquad\text{(C) }\frac{4}{3}\qquad\text{(D) }\frac{7}{3}\qquad\text{(E) }\frac{13}{5}$

Define three sequences $a_n,b_n,c_n$ as $a_0=b_0=c_0=1$ and \begin{align*} a_{n+1}&=a_n+3b_n+3c_n \\ b_{n+1}&=a_n+b_n+3c_n \\ c_{n+1}&=a_n+b_n+c_n \end{align*} for $n\geq0$. Let $A,B,C$ be the remainders when $a_{13^4},b_{13^4},c_{13^4}$ are divided by $13$. Find the ordered triple $\left(A,B,C\right)$. [i]2019 CCA Math Bonanza Team Round #10[/i]

A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$? [hide="Clarifications"] [list] [*] $S$ is the ``set of 10 distinct integers'' from the first sentence.[/list][/hide] [i]Ray Li[/i]

Given an triangle $ABC$ isosceles at the vertex $A$, let $P$ and $Q$ be the touchpoints with $AB$ and $AC$, respectively with the circle $T$, which is tangent to both and is internally tangent to the circumcircle of $ABC$. Let $R$ and $S$ be the points of the circumscribed circle of $ABC$ such that $AP = AR = AS$ . Prove that $RS$ is tangent to $T$ .

Let $ABC$ be a triangle and let $D$ and $E$ be points on sides $AB$ and $AC$, respectively, such that $DE \parallel BC$. Let $P$ be any point interior to triangle $ADE$, and let $F$ and $G$ be the intersections of $DE$ with the lines $BP$ and $CP$, respectively. Let $Q$ be the second intersection point of the circumcircles of triangles $PDG$ and $PFE$. Prove that the points $A,P,$ and $Q$ are collinear.

For integer $n > 1$, consider $n$ cube polynomials $P_1(x), ..., P_n(x)$ such that each polynomial has $3$ distinct real roots. Denote $S$ as the set of roots of following equation $P_1(x)P_2(x)P_3(x)... P_n(x) = 0$. It is also known that for each $1 \le i < j \le n, P_i(x)P_j(x) = 0$ has $5$ distinct real roots. 1. Prove that if for each $a, b \in S$, there is exactly one $i \in\{1,2, 3,..., n\}$ such that $P_i(a) = P_i(b) = 0$ then $n = 7$. 2. Prove that if $n > 7$ then $|S| = 2n + 1$.

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

In the grid shown, fill in each empty space with a number, such that after the grid is completely filled in, the number in each space is equal to the smallest positive integer that does not appear in any of the touching spaces. (A pair of spaces is considered to touch if they both share a vertex.) You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(.5cm); int count=0; real s3=1.73205081; pair[] points={(-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3)}; void draw_hexagon(pair p) { draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30))); } void add_number() { draw_hexagon(points[count]); count+=1; } void add_number(int n) { label((string)n, points[count]); add_number(); } add_number(4); add_number(); add_number(); add_number(1); add_number(1); add_number(); add_number(1); add_number(); add_number(3); add_number(); add_number(2); add_number(); add_number(); add_number(6); add_number(); add_number(); add_number(); add_number(); add_number(7); add_number(); add_number(); add_number(); add_number(); add_number(3); add_number(); add_number(); add_number(5); add_number(); add_number(2); add_number(); add_number(4); add_number(); add_number(3); add_number(3); add_number(); add_number(); add_number(2);[/asy]

All empty white triangles in figure are to be filled with integers such that for each gray triangle the three numbers in the white neighboring triangles sum to a multiple of $5$. The lower left and the lower right white triangle are already filled with the numbers $12$ and $3$, respectively. Find all integers that can occur in the uppermost white triangle. (G. Woeginger, Eindhoven, The Netherlands) [img]https://cdn.artofproblemsolving.com/attachments/8/a/764732f5debbd58a147e9067e83ba8d31f7ee9.png[/img]

What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.

Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?

The numbers $1, 2, 3, ..., 1000$ are written on the board. Two people take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by three, then the one who made the first move wins. if not, then his partner. Which one will win if played correctly?

In circle $\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_1$ and $M_2$ respectively, such that $PM_1=15$ and $PM_2=20$. Line $M_1M_2$ intersects $\omega$ at points $A$ and $B$, with $M_1$ between $A$ and $M_2$. Compute the largest possible value of $BM_2-AM_1$.

Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that: i) For all real $x$ we have $F(x)\le G(x)\le H(x)$. ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)\quad\text{for}\ i=1,2,3,\ldots ,n$. iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$. Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$. (b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$. (c) $|A| = |B| = |C| = 4$. Here, $|S|$ denotes the number of elements in the set $S$.

How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?

The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations: \begin{align*}2x+3y+3z&=8,\\3x+2y+3z&=808,\\3x+3y+2z&=80808.\end{align*} Their goal is not to find a solution $(x,y,z)$ to the system, but instead to compute the sum of the variables. Find the value of $x+y+z$.

Let $ABC$ be a triangle with $I$ as incenter.The incircle touches $BC$ at $D$.Let $D'$ be the antipode of $D$ on the incircle.Make a tangent at $D'$ to incircle.Let it meet $(ABC)$ at $X,Y$ respectively.Let the other tangent from $X$ meet the other tangent from $Y$ at $Z$.Prove that $(ZBD)$ meets $IB$ at the midpoint of $IB$

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

A convex quadrilateral $ABCD$ is given where $\angle DAB =\angle ABC = 120^o$ and $CD = 3$,$BC = 2$, $AB = 1$. Calculate the length of segment $AD$.

$$\int_{-\infty}^\infty \frac{\sin(\pi x)}{x(1+x^2)}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

Let $n$ be a positive integer and let vectors $v_1$, $v_2$, $\ldots$, $v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i$th minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n$th minute there exists no point which is occupied by the flea with greater probability than the origin. [i]Proposed by Péter Pál Pach, Budapest[/i]

Compute \[\sum_{n_{60}=0}^2\sum_{n_{59}=0}^{n_{60}}\cdots\sum_{n_2=0}^{n_3}\sum_{n_1=0}^{n_2}\sum_{n_0=0}^{n_1}1.\]