[u]Round 5[/u] [b]p13.[/b] Vincent the Bug is at the vertex $A$ of square $ABCD$. Each second, he moves to an adjacent vertex with equal probability. The probability that Vincent is again on vertex $A$ after $4$ seconds is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p14.[/b] Let $ABC$ be a triangle with $AB = 2$, $AC = 3$, and $\angle BAC = 60^o$. Let $P$ be a point inside the triangle such that $BP = 1$ and $CP =\sqrt3$, let $x$ equal the area of $APC$. Compute $16x^2$. [b]p15.[/b] Let $n$ be the number of multiples of$ 3$ between $2^{2020}$ and $2^{2021}$. When $n$ is written in base two, how many digits in this representation are $1$? [u]Round 6[/u] [b]p16.[/b] Let $f(n)$ be the least positive integer with exactly n positive integer divisors. Find $\frac{f(200)}{f(50)}$ . [b]p17.[/b] The five points $A, B, C, D$, and $E$ lie in a plane. Vincent the Bug starts at point $A$ and, each minute, chooses a different point uniformly at random and crawls to it. Then the probability that Vincent is back at $A$ after $5$ minutes can be expressed as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [b]p18.[/b] A circle is divided in the following way. First, four evenly spaced points $A, B, C, D$ are marked on its perimeter. Point $P$ is chosen inside the circle and the circle is cut along the rays $PA$, $PB$, $PC$, $PD$ into four pieces. The piece bounded by $PA$, $PB$, and minor arc $AB$ of the circle has area equal to one fifth of the area of the circle, and the piece bounded by $PB$, $PC$, and minor arc $BC$ has area equal to one third of the area of the circle. Suppose that the ratio between the area of the second largest piece and the area of the circle is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p + q$. [u]Round 7 [/u] [b]p19.[/b] There exists an integer $n$ such that $|2^n - 5^{50}|$ is minimized. Compute $n$. [b]p20.[/b] For nonnegative integers $a = \overline{a_na_{n-1} ... a_2a_1}$, $b = \overline{b_mb_{m-1} ... b_2b_1}$, define their distance to be $$d(a, b) = \overline{|a_{\max\,\,(m,n)} - b_{\max\,\,(m,n)}||a_{\max\,\,(m,n)-1} - b_{\max\,\,(m,n)-1}|...|a_1 - b_1|}$$ where $a_k = 0$ if $k > n$, $b_k = 0$ if $k > m$. For example, $d(12321, 5067) = 13346$. For how many nonnegative integers $n$ is $d(2021, n) + d(12345, n)$ minimized? [b]p21.[/b] Let $ABCDE$ be a regular pentagon and let $P$ be a point outside the pentagon such that $\angle PEA = 6^o$ and $\angle PDC = 78^o$. Find the degree-measure of $\angle PBD$. [u]Round 8[/u] [b]p22.[/b] What is the least positive integer $n$ such that $\sqrt{n + 3} -\sqrt{n} < 0.02$ ? [b]p23.[/b] What is the greatest prime divisor of $20^4 + 21 \cdot 23 - 6$? [b]p24.[/b] Let $ABCD$ be a parallelogram and let $M$ be the midpoint of $AC$. Suppose the circumcircle of triangle $ABM$ intersects $BC$ again at $E$. Given that $AB = 5\sqrt2$, $AM = 5$, $\angle BAC$ is acute, and the area of $ABCD$ is $70$, what is the length of $DE$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2949414p26408213]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$

Find the smallest value of a function $$y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.$$

A sequence $\{z_n\}$ satisfies that for any positive integer $i,$ $z_i\in\{0,1,\cdots,9\}$ and $z_i\equiv i-1 \pmod {10}.$ Suppose there is $2021$ non-negative reals $x_1,x_2,\cdots,x_{2021}$ such that for $k=1,2,\cdots,2021,$ $$\sum_{i=1}^kx_i\geq\sum_{i=1}^kz_i,\sum_{i=1}^kx_i\leq\sum_{i=1}^kz_i+\sum_{j=1}^{10}\dfrac{10-j}{50}z_{k+j}.$$ Determine the least possible value of $\sum_{i=1}^{2021}x_i^2.$

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (where $\mathbb{Z}$ is the set of all integers) such that \[ 2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}. \]

For any positive integer $x$, we set $$ g(x) = \text{ largest odd divisor of } x, $$ $$ f(x) = \begin{cases} \frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\ 2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.} \end{cases} $$ Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that $$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$ [hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

What conditions should real numbers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ meet in order for a polynomial of second degree $$ax^2 + 2bxy + cy^2 + 2dx + 2ey + f$$ was the product of two first degree polynomials with real coefficients ?

problem 1 :A sequence is defined by$ x_1 = 1, x_2 = 4$ and $ x_{n+2} = 4x_{n+1} -x_n$ for $n \geq 1$. Find all natural numbers $m$ such that the number $3x_n^2 + m$ is a perfect square for all natural numbers $n$

For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$ Find the value of the expression $x^2 + y^2+ z^2$.

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

Solve the equation (for positive $x$) $$x^x=\frac{1}{\sqrt2}$$

$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.

Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$? [i]Adam Ardeishar and Michael Ren[/i]

[b]p1.[/b] On SeaBay, green herring costs $\$2.50$ per pound, blue herring costs $\$4.00$ per pound, and red herring costs $\$5,85$ per pound. What must Farmer James pay for $12$ pounds of green herring and $7$ pounds of blue herring, in dollars? [b]p2.[/b] A triangle has side lengths $3$, $4$, and $6$. A second triangle, similar to the first one, has one side of length $12$. Find the sum of all possible lengths of the second triangle's longest side. [b]p3.[/b] Hen Hao runs two laps around a track. Her overall average speed for the two laps was $20\%$ slower than her average speed for just the first lap. What is the ratio of Hen Hao's average speed in the first lap to her average speed in the second lap? [b]p4.[/b] Square $ABCD$ has side length $2$. Circle $\omega$ is centered at $A$ with radius $2$, and intersects line $AD$ at distinct points $D$ and $E$. Let $X$ be the intersection of segments $EC$ and $AB$, and let $Y$ be the intersection of the minor arc $DB$ with segment $EC$. Compute the length of $XY$ . [b]p5.[/b] Hen Hao rolls $4$ tetrahedral dice with faces labeled $1$, $2$, $3$, and $4$, and adds up the numbers on the faces facing down. Find the probability that she ends up with a sum that is a perfect square. [b]p6.[/b] Let $N \ge 11$ be a positive integer. In the Eggs-Eater Lottery, Farmer James needs to choose an (unordered) group of six different integers from $1$ to $N$, inclusive. Later, during the live drawing, another group of six numbers from $1$ to $N$ will be randomly chosen as winning numbers. Farmer James notices that the probability he will choose exactly zero winning numbers is the same as the probability that he will choose exactly one winning number. What must be the value of $N$? [b]p7.[/b] An egg plant is a hollow cylinder of negligible thickness with radius $2$ and height $h$. Inside the egg plant, there is enough space for four solid spherical eggs of radius $1$. What is the minimum possible value for $h$? [b]p8.[/b] Let $a_1, a_2, a_3, ...$ be a geometric sequence of positive reals such that $a_1 < 1$ and $(a_{20})^{20} = (a_{18})^{18}$. What is the smallest positive integer n such that the product $a_1a_2a_3...a_n$ is greater than $1$? [b]p9.[/b] In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$. [b]p10.[/b] Farmer James places the numbers $1, 2,..., 9$ in a $3\times 3$ grid such that each number appears exactly once in the grid. Let $x_i$ be the product of the numbers in row $i$, and $y_i$ be the product of the numbers in column $i$. Given that the unordered sets $\{x_1, x_2, x_3\}$ and $\{y_1, y_2, y_3\}$ are the same, how many possible arrangements could Farmer James have made? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$

Let $S$ be a set of positive real numbers with five elements such that for any distinct $a,~ b,~ c$ in $S$, the number $ab + bc + ca$ is rational. Prove that for any $a$ and $b$ in $S$, $\tfrac{a}{b}$ is a rational number.

[b]p1.[/b] What is the maximal number of pieces of two shapes, [img]https://cdn.artofproblemsolving.com/attachments/a/5/6c567cf6a04b0aa9e998dbae3803b6eeb24a35.png[/img] and [img]https://cdn.artofproblemsolving.com/attachments/8/a/7a7754d0f2517c93c5bb931fb7b5ae8f5e3217.png[/img], that can be used to tile a $7\times 7$ square? [b]p2.[/b] Six shooters participate in a shooting competition. Every participant has $5$ shots. Each shot adds from $1$ to $10$ points to shooter’s score. Every person can score totally for all five shots from $5$ to $50$ points. Each participant gets $7$ points for at least one of his shots. The scores of all participants are different. We enumerate the shooters $1$ to $6$ according to their scores, the person with maximal score obtains number $1$, the next one obtains number $2$, the person with minimal score obtains number $6$. What score does obtain the participant number $3$? The total number of all obtained points is $264$. [b]p2.[/b] There are exactly $n$ students in a high school. Girls send messages to boys. The first girl sent messages to $5$ boys, the second to $7$ boys, the third to $6$ boys, the fourth to $8$ boys, the fifth to $7$ boys, the sixth to $9$ boys, the seventh to $8$, etc. The last girl sent messages to all the boys. Prove that $n$ is divisible by $3$. [b]p4.[/b] In what minimal number of triangles can one cut a $25 \times 12$ rectangle in such a way that one can tile by these triangles a $20 \times 15$ rectangle. [b]p5.[/b] There are $2014$ stones in a pile. Two players play the following game. First, player $A$ takes some number of stones (from $1$ to $30$) from the pile, then player B takes $1$ or $2$ stones, then player $A$ takes $2$ or $3$ stones, then player $B$ takes $3$ or $4$ stones, then player A takes $4$ or $5$ stones, etc. The player who gets the last stone is the winner. If no player gets the last stone (there is at least one stone in the pile but the next move is not allowed) then the game results in a draw. Who wins the game using the right strategy? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given is a positive integer $n \geq 2$. Determine the maximum value of the sum of natural numbers $k_1,k_2,...,k_n$ satisfying the condition: $k_1^3+k_2^3+ \dots +k_n^3 \leq 7n$.

[b]p1. [/b] What is the maximum number of $T$-shaped polyominos (shown below) that we can put into a $6 \times 6$ grid without any overlaps. The blocks can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/7/6/468fd9b81e9115a4a98e4cbf6dedf47ce8349e.png[/img] [b]p2.[/b] In triangle $\vartriangle ABC$, $\angle A = 30^o$. $D$ is a point on $AB$ such that $CD \perp AB$. $E$ is a point on $AC$ such that $BE \perp AC$. What is the value of $\frac{DE}{BC}$ ? [b]p3.[/b] Given that f(x) is a polynomial such that $2f(x) + f(1 - x) = x^2$. Find the sum of squares of the coefficients of $f(x)$. [b]p4. [/b] For each positive integer $n$, there exists a unique positive integer an such that $a^2_n \le n < (a_n + 1)^2$. Given that $n = 15m^2$ , where $m$ is a positive integer greater than $1$. Find the minimum possible value of $n - a^2_n$. [b]p5.[/b] What are the last two digits of $\lfloor (\sqrt5 + 2)^{2016}\rfloor$ ? Note $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p6.[/b] Let $f$ be a function that satisfies $f(2^a3^b)) = 3a+ 5b$. What is the largest value of f over all numbers of the form $n = 2^a3^b$ where $n \le 10000$ and $a, b$ are nonnegative integers. [b]p7.[/b] Find a multiple of $21$ such that it has six more divisors of the form $4m + 1$ than divisors of the form $4n + 3$ where m, n are integers. You can keep the number in its prime factorization form. [b]p8.[/b] Find $$\sum^{100}_{i=0} \lfloor i^{3/2} \rfloor +\sum^{1000}_{j=0} \lfloor j^{2/3} \rfloor$$ where $\lfloor x \rfloor$ is the largest integer less or equal to x. [b]p9. [/b] Let $A, B$ be two randomly chosen subsets of $\{1, 2, . . . 10\}$. What is the probability that one of the two subsets contains the other? [b]p10.[/b] We want to pick $5$-person teams from a total of $m$ people such that: 1. Any two teams must share exactly one member. 2. For every pair of people, there is a team in which they are teammates. How many teams are there? (Hint: $m$ is determined by these conditions). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].