Prove that the number of orientations of a connected $3$-regular graph on $2n$ vertices where the number of vertices with indegree $0$ and outdegree $0$ are equal, is exactly $2^{n+1}$ $ {2n} \choose {n}$.

Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$: \[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]

We say that we extend a finite sequence of positive integers $(a_1,\dotsc,a_n)$ if we replace it by \[(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n)\] i.e., each element $k$ of the original sequence is replaced by $1,2,\dotsc,k$. Géza takes the sequence $(1,2,\dotsc,9)$ and he extends it $2017$ times. Then he chooses randomly one element of the resulting sequence. What is the probability that the chosen element is $1$?

For this exercise, $ \{\} $ denotes the fractional part. [b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $ [b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $ [i]Traian Preda[/i]

Given a $6\times 6$ square consisting of unit squares, denote its rows and columns from $1$ to $6$. Figure [i]p-horse[/i] can move from square $(x; y)$ to $(x’; y’)$ if and only if both $x + x’$ and $y + y’$ are primes. At the start the [i]p-horse[/i] is located in one of the unit squares. $a)$ Can the [i]p-horse[/i] visit every unit square exactly once? $b$) Can the [i]p-horse[/i] visit every unit square exactly once and with the last move return to the initial starting position?

Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$. Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$, define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way: (a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$. (b) $B_{i+1}\neq C_i$ is the intersection of line $A_iC_i$ with the circumcircle of $O_iA_{i+1}C_i$. (c) $C_{i+1}\neq B_i$ is the intersection of line $A_iB_i$ with the circumcircle of $O_iA_{i+1}B_i$. Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \] Note: $[K]$ denotes the area of $K$. [i]Proposed by Yang Liu[/i]

Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that \[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

Define a function $f$ on the positive integers as follows: $f(n) = m$, where $m$ is the least positive integer such that $n$ is a factor of $m^2$. Find the smallest integer $M$ such that $\sqrt{M}$ is both a product of prime numbers, of which there are at least $3$, and a factor of $$\sum_{ d|M} f(d),$$ the sum of $f(d)$ for all positive integers $d$ that divide $M$.

Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$ coincide, then it must be a square.

For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy $$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$ [i]Proposed by Kyle Lee[/i]

Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$. Proposed by: P.Puchkov,E.Utkin

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

In $\triangle{ABC}$ with an obtuse angle at $A$, let $D$ be the foot of the $A$ altitude and $E$ be the foot of the $B$ altitude. If $AC+CD=DB$ and $BC-AE=EC$, compute $\angle{A}$ in degrees. [i]2020 CCA Math Bonanza Team Round #10[/i]

[b]p1.[/b] Suppose that Yunseo wants to order a pizza that is cut into $4$ identical slices. For each slice, there are $2$ toppings to choose from: pineapples and apples. Each slice must have exactly one topping. How many distinct pizzas can Yunseo order? Pizzas that can be obtained by rotating one pizza are considered the same. [b]p2.[/b] How many triples of distinct positive integers $(E, M, C)$ are there such that $E = MC^2$ and $E \le 50$? [b]p3.[/b] Given that the cubic polynomial $p(x)$ has leading coefficient $1$ and satisfies $p(0) = 0$, $p(1) = 1$, and $p(2) = 2$. Find $p(3)$. [b]p4.[/b] Olaf asks Anna to guess a two-digit number and tells her that it’s a multiple of $7$ with two distinct digits. Anna makes her first guess. Olaf says one digit is right but in the wrong place. Anna adjusts her guess based on Olaf’s comment, but Olaf answers with the same comment again. Anna now knows what the number is. What is the sum of all the numbers that Olaf could have picked? [b]p5.[/b] Vincent the Bug draws all the diagonals of a regular hexagon with area $720$, splitting it into many pieces. Compute the area of the smallest piece. [b]p6.[/b] Given that $y - \frac{1}{y} = 7 + \frac{1}{7}$, compute the least integer greater than $y^4 + \frac{1}{y^4}$. [b]p7.[/b] At $9:00$ A.M., Joe sees three clouds in the sky. Each hour afterwards, a new cloud appears in the sky, while each old cloud has a $40\%$ chance of disappearing. Given that the expected number of clouds that Joe will see right after $1:00$ P.M. can be written in the form $p/q$ , where $p$ and $q$ are relatively prime positive integers, what is $p + q$? [b]p8.[/b] Compute the unique three-digit integer with the largest number of divisors. [b]p9.[/b] Jo has a collection of $101$ books, which she reads one each evening for $101$ evenings in a predetermined order. In the morning of each day that Jo reads a book, Amy chooses a random book from Jo’s collection and burns one page in it. What is the expected number of pages that Jo misses? [b]p10.[/b] Given that $x, y, z$ are positive real numbers satisfying $2x + y = 14 - xy$, $3y + 2z = 30 - yz$, and $z + 3x = 69 - zx$, the expression $x + y + z$ can be written as $p\sqrt{q} - r$, where $p, q, r$ are positive integers and $q$ is not divisible by the square of any prime. Compute $p + q + r$. [b]p11.[/b] In rectangle $TRIG$, points $A$ and $L$ lie on sides $TG$ and $TR$ respectively such that $TA = AG$ and $TL = 2LR$. Diagonal $GR$ intersects segments $IL$ and $IA$ at $B$ and $E$ respectively. Suppose that the area of the convex pentagon with vertices $TABLE$ is equal to $21$. What is the area of $TRIG$? [b]p12.[/b] Call a number nice if it can be written in the form $2^m \cdot 3^n$, where $m$ and $n$ are nonnegative integers. Vincent the Bug fills in a $3$ by $3$ grid with distinct nice numbers, such that the product of the numbers in each row and each column are the same. What is the smallest possible value of the largest number Vincent wrote? [b]p13.[/b] Let $s(n)$ denote the sum of digits of positive integer $n$ and define $f(n) = s(202n) - s(22n)$. Given that $M$ is the greatest possible value of $f(n)$ for $0 < n < 350$ and $N$ is the least value such that $f(N) = M$, compute $M + N$. [b]p14.[/b] In triangle $ABC$, let M be the midpoint of $BC$ and let $E, F$ be points on $AB, AC$, respectively, such that $\angle MEF = 30^o$ and $\angle MFE = 60^o$. Given that $\angle A = 60^o$, $AE = 10$, and $EB = 6$,compute $AB + AC$. [b]p15.[/b] A unit cube moves on top of a $6 \times 6$ checkerboard whose squares are unit squares. Beginning in the bottom left corner, the cube is allowed to roll up or right, rolling about its bottom edges to travel from square to square, until it reaches the top right corner. Given that the side of the cube facing upwards in the beginning is also facing upwards after the cube reaches the top right corner, how many total paths are possible? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b,c$ be positive real numbers. Prove that \[\frac{a}b+\frac{b}c+\frac{c}a \geq \frac{c+a}{c+b}+\frac{a+b}{a+c}+\frac{b+c}{b+a}\]

Do there exist real numbers $a,b,c$ such that the system of equations \begin{align*} x+y+z&=a\\ x^2+y^2+z^2&=b\\ x^4+y^4+z^4&=c \end{align*} has infinitely many real solutions $(x,y,z)$?

$ABCD$ is a square side $1$. $P$ and $Q$ lie on the side $AB$ and $R$ lies on the side $CD$. What are the possible values for the circumradius of $PQR$?

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

Anna took a number \(N\), which is written in base 10 and has fewer than 9 digits, and duplicated it by writing another \(N\) to its left, creating a new number with twice as many digits. Bob computed the sum of all integers from 1 to \(N\). It turned out that Anna's new number is 7 times as large as the sum computed by Bob. Determine \(N\). [i]Proposed by Bachana Kutsia, Georgia [/i]

An integer $n>1$ , whose positive divisors are $1=d_1<d_2< \cdots <d_k=n$, is called $\textit{southern}$ if all the numbers $d_2-d_1, d_3- d_2 , \cdots, d_k-d_{k-1}$ are divisors of $n$. a) Find a positive integer that is $\textit{not southern}$ and has exactly $2022$ positive divisors that are $\textit{southern}$. b) Show that there are infinitely many positive integers that are $\textit{not southern}$ and have exactly $2022$ positive divisors that are $\textit{southern}$.

For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?

Given is a rhombus $ABCD$ of side 1. On the sides $BC$ and $CD$ we are given the points $M$ and $N$ respectively, such that $MC+CN+MN=2$ and $2\angle MAN = \angle BAD$. Find the measures of the angles of the rhombus. [i]Cristinel Mortici[/i]

On a table, cards numbered $1, 2, \ldots , 200$ are laid out in a row in some order, and a line is drawn on the table between some two of them. It is allowed to swap two adjacent cards if the number on the left is greater than the number on the right. After a few such moves, the cards were arranged in ascending order. Prove we have swapped pairs of cards separated by the line no more than 1884 times.