Bob sends a secret message to Alice using her RSA public key $n = 400000001.$ Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n.$ Eve knows that $n = pq,$ a product of two prime factors. Find $p$ and $q.$

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$. [i]Proposed by Athanasios Kontogeorgis, Greece[/i]

In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$. Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$. The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$. Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.

If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals A. -12 B. 24 C. 35 D. -61 E. -63

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

An infinite array of rational numbers $G(d, n)$ is defined for integers $d$ and $ n$ with $1\leq d \leq n$ as follows: $$G(1, n)= \frac{1}{n}, \;\;\; G(d,n)= \frac{d}{n} \sum_{i=d}^{n} G(d-1, i-1) \; \text{for} \; d>1.$$ For $1 < d < p$ and $p$ prime, prove that $G(d, p)$ is expressible as a quotient $s\slash t$ of integers $s$ and $t$ with $t$ not divisible by $p.$

A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$, $OB$ and $OC$ are equal to $15$,$12$ and $20$, respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle. Find the value of the angle $BAC$.

A male volcano is in the shape of a hollow cone with the point side up, but with everything above a height of 6 meters removed. The resulting shape has a bottom radius of 10 meters and a top radius of 7 meters, with a height of 6 meters. He sat above his bay, watching all the couples play. His lava grew and grew until he was half full of lava. Then, he erupted, lowering the height of the lava to 2 meters. What fraction of the lava remained in the volcano? [i]Proposed by Matthew Weiss

[b]2.[/b] Show that the series $\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}$ is convergent for every positive integer N and any real numbers a and b. [b](S. 25)[/b]

In a forest each of $n$ animals ($n\ge 3$) lives in its own cave, and there is exactly one separate path between any two of these caves. Before the election for King of the Forest some of the animals make an election campaign. Each campaign-making animal visits each of the other caves exactly once, uses only the paths for moving from cave to cave, never turns from one path to another between the caves and returns to its own cave in the end of its campaign. It is also known that no path between two caves is used by more than one campaign-making animal. a) Prove that for any prime $n$, the maximum possible number of campaign-making animals is $\frac{n-1}{2}$. b) Find the maximum number of campaign-making animals for $n=9$.

Let m and n be positive integers. Alice wishes to walk from the point $(0, 0)$ to the point $(m,n)$ in increments of $(1, 0)$ and $(0, 1)$, and Bob wishes to walk from the point $(0,1)$ to the point $(m, n + 1)$ in increments of$ (1, 0)$ and $(0,1)$. Find (with proof) the number of ways for Alice and Bob to get to their destinations if their paths never pass through the same point (even at different times).

Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]

Alice, Bob, and Catherine decide to have a race. Alice runs at a speed of $3$ feet per second, and Bob runs at a speed of $5$ feet per second. In the end, Bob finishes the same amount of time before Catherine as Catherine finishes before Alice. What was Catherine's speed, in feet per second? $\textbf{(A) }\dfrac{15}4\qquad\textbf{(B) }4\qquad\textbf{(C) }\dfrac{17}4\qquad\textbf{(D) }\dfrac92\qquad\textbf{(E) }\text{impossible to determine}$

Find the sum of all integers $n$ satisfying the following inequality: \[\frac{1}{4}<\sin\frac{\pi}{n}<\frac{1}{3}.\]

Let be two real numbers $ a>0,b. $ Calculate the primitive of the function $ 0<x\mapsto\frac{bx-1}{e^{bx}+ax} . $ [i]Dan Negulescu[/i]

In an art museum, $n$ paintings are exhibited, where $n \geq 33.$ In total, $15$ colors are used for these paintings such that any two paintings have at least one common color, and no two paintings have exactly the same colors. Determine all possible values of $n \geq 33$ such that regardless of how we color the paintings with the given properties, we can choose four distinct paintings, which we can label as $T_1, T_2, T_3,$ and $T_4,$ such that any color that is used in both $T_1$ and $T_2$ can also be found in either $T_3$ or $T_4$.

Ayala and Barvaz play a game: Ayala initially gives Barvaz two $100\times100$ tables of positive integers, such that the product of numbers in each table is the same. In one move, Barvaz may choose a row or column in one of the tables, and change the numbers in it (to some positive integers), as long as the total product remains the same. Barvaz wins if after $N$ such moves, he manages to make the two tables equal to each other, and otherwise Ayala wins. a. For which values of $N$ does Barvaz have a winning strategy? b. For which values of $N$ does Barvaz have a winning strategy, if all numbers in Ayalah’s tables must be powers of $2$?

Is there a function $f(x)$ defined and continuous on $R$ such that: a) $f(f(x)) = 1 + 2x$ ? b) $f(f(x)) = 1 - 2x $?

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur. (I.Bogdanov)

It is given a circle with center $O$ and radius $r$. $AB$ and $MN$ are two diameters. The lines $MB$ and $NB$ are tangent to the circle at the points $M'$ and $N'$ and intersect at point $A$. $M''$ and $N''$ are the midpoints of the segments $AM'$ and $AN'$. Prove that: (a) the points $M,N,N',M'$ are concyclic. (b) the heights of the triangle $M''N''B$ intersect in the midpoint of the radius $OA$.

2) A mass $m$ hangs from a massless spring connected to the roof of a box of mass $M$. When the box is held stationary, the mass–spring system oscillates vertically with angular frequency $\omega$. If the box is dropped and falls freely under gravity, how will the angular frequency change? A) $\omega$ will be unchanged B) $\omega$ will increase C) $\omega$ will decrease D) Oscillations are impossible under these conditions. E) $\omega$ will either increase or decrease depending on the values of $M$ and $m$.

A person starting with $64$ cents and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is: $\textbf{(A)}\ \text{a loss of } 27 \qquad \textbf{(B)}\ \text{a gain of }27 \qquad \textbf{(C)}\ \text{a loss of }37 \qquad$ $ \textbf{(D)}\ \text{neither a gain nor a loss} \qquad \textbf{(E)}\ \text{a gain or a loss depending upon the order in which the wins and losses occur}$ Note: Due to the lack of $\LaTeX$ packages, the numbers in the answer choices are in cents ¢

Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$ Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$

A square is inscribed in a circle of radius $6$. A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, what is $n$? [asy] size(5 cm); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw(circle((1,1),1.41)); draw(arc((0,0),2,0,90));[/asy] [i]2021 CCA Math Bonanza Lightning Round #1.2[/i]