[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].

Each pair of three circles have the common chords $AA_1, BB_1$ and $CC_1{}$ such that lines $AB{}$ and $A_1B_1$ intersect in point $M{}$, $BC$ and $B_1C_1$ intersect in point $N{}$, $CA{}$ and $C_1A_1$ intersect in point $P{}$. Prove that points $M, N$ and $P$ are collinear.

Determine all positive real numbers $x$ and $y$ satisfying the equation \[x+y+\frac{1}{x}+\frac{1}{y}+4=2\cdot (\sqrt{2x+1}+\sqrt{2y+1})\]

Given a triangle $ABC$ with $\angle ACB = 120^{\circ}.$ A point $L$ is marked in the side $AB$ such that $CL$ bisects $\angle ACB.$ Points $N$ and $K$ are chosen in the sides $AC$ and $BC $ such that $CK+CN=CL.$ Prove that the triangle $KLN$ is equilateral.

Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals. Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.

Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define \[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.

Let $\Omega$ and $\Gamma$ two circumferences such that $\Omega$ is in interior of $\Gamma$. Let $P$ a point on $\Gamma$. Define points $A$ and $B$ distinct of $P$ on $\Gamma$ such that $PA$ and $PB$ are tangentes to $\Omega$. Prove that when $P$ varies on $\Gamma$, the line $AB$ is tangent to a fixed circunference.

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

Let $\omega$ be the circumcircle of an actue triangle $ABC$ and let $H$ be the feet of aliitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$. The lines $BM$ and $CN$ intersect each other at $G$ and intersect $\omega$ at $P$ and $Q$ respectively. The circles $(HMG)$ and $(HNG)$ intersect the segments $HP$ and $HQ$ again at $R$ and $S$ respectively. Prove that $PQ\parallel RS$.

Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. What is $\cos({\angle CMD})$? $\textbf{(A)} ~\frac{1}{4} \qquad\textbf{(B)} ~\frac{1}{3} \qquad\textbf{(C)} ~\frac{2}{5} \qquad\textbf{(D)} ~\frac{1}{2} \qquad\textbf{(E)} ~\frac{\sqrt{3}}{2} $

We call a collection of weights (each weighing an integer value) basic if their total weight equals $200$ and each object of integer weight not greater than $200$ can be balanced exactly with a uniquely determined set of weights from the collection. (Uniquely means that we are not concerned with order or which weights of equalc value are chosen to balance against a particular object, if in fact there is a choice.) (a) Find an example of a basic collection other than the collection of $200$ weights each of value $1$. (b) How many different basic collections are there? (D. Fomin, Leningrad)

A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid. [asy] size(150); defaultpen(linewidth(0.8)); draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1)); draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4")); draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4")); label("$5$",(0,5/2),W); label("$8$",(4,0),S); [/asy]

High schoolers chew a lot of gum. At the supermarket, $15$ packs of $14$ sticks of gum costs $\$10$. If $1400$ high schoolers chew $3$ sticks of gum per day, find the total number of dollars spent by these high schoolers on gum per week.

Find all monic polynomials $P(x),Q(x)$ with integer coefficients such that $Q(0) =0$ and $P(Q(x)) = (x-1)(x-2)...(x-15)$.

Let $a_1, a_2, ..., a_n$ be positive real numbers. Furthermore, let $S_n$ denote the set of all permutations of set $\{1, 2, ..., n\}$. Prove that $$\sum_{\pi \in S_n} \frac{1}{a_{\pi(1)}(a_{\pi(1)}+a_{\pi(2)})...(a_{\pi(1)}+a_{\pi(2)}+...+a_{\pi(n)})}=\frac{1}{a_1 a_2 ... a_n}$$

Alice and the Mad Hatter are playing a game. At the start of the game, three $2024$’s are written on the blackboard. Then, Alice and the Mad Hatter alternate turns, with the Mad Hatter starting. On the Mad Hatter’s turn, he must pick one of the numbers on the blackboard and increase it by $1$. On Alice’s turn, she must: - pick one of the numbers on the blackboard and decrease it by 1, and then - replace the two numbers $a$ and $b$ on the blackboard which were not chosen by the Mad Hatter on the previous turn with $\sqrt{ab}$. Alice wins if, on the start of her turn, any of the three numbers are less than $1$. Can the Mad Hatter prevent Alice from winning?

A dart is thrown at a square dartboard of side length $2$ so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance $1$ of a corner?

The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that \[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\] for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that \[\varphi(x,y,z) = h(x+y+z)\] for all real numbers $x,y$ and $z.$

4) find all polynom with coeffs a permutation of $[1,...,n]$ and all roots rational

We consider the set \begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad \nu \geq 2 \end{align*} and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to \begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*} We also consider the $\nu-$tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$th roots of $-1$, where \begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \qquad \mu =0,1, \ldots , \nu -1 \end{align*} Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$, we obtain the element \begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*} Determine [list=i] [*] the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) $ have measures less than or equal to $1$ [*] for $\nu =4$, the minimal value of $\kappa \in \mathbb{N}$, for which \begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} \qquad \qquad 1 \le i \le 4 \end{align*}

Let $x_1, x_2, x_3, y_1, y_2, y_3$ be real numbers in $[-1, 1]$. Find the maximum value of \[(x_1y_2-x_2y_1)(x_2y_3-x_3y_2)(x_3y_1-x_1y_3).\]

Let $ABCD$ be a trapezoid, where $AD\parallel BC$, $BC<AD$, and $AB\cap DC=T$. A circle $k_1$ is inscribed in $\Delta BCT$ and a circle $k_2$ is an excircle for $\Delta ADT$ which is tangent to $AD$ (opposite to $T$). Prove that the tangent line to $k_1$ through $D$, different than $DC$, is parallel to the tangent line to $k_2$ through $B$, different than $BA$.

Let $ABCD$ be the cyclic quadrilateral. Suppose that there exists some line $l$ parallel to $BD$ which is tangent to the inscribed circles of triangles $ABC, CDA$. Show that $l$ passes through the incenter of $BCD$ or through the incenter of $DAB$. [i](Proposed by Fedir Yudin)[/i]

In the rectangular coordinate system every point with integer coordinates is called laticeal point. Let $P_n(n, n + 5)$ be a laticeal point and denote by $f(n)$ the number of laticeal points on the open segment $(OP_n)$, where the point $0(0,0)$ is the coordinates system origine. Calculate the number $f(1) +f(2) + f(3) + ...+ f(2002) + f(2003)$.

Let $ m,n > 1$ are integers which satisfy $ n|4^m \minus{} 1$ and $ 2^m|n \minus{} 1$. Is it a must that $ n \equal{} 2^{m} \plus{} 1$?