On her blackboard, Alice has written $n$ integers strictly greater than $1$. Then, she can, as often as she likes, erase two numbers $a$ and $b$ such that $a \neq b$, and replace them with $q$ and $q^2$, where $q$ is the product of the prime factors of $ab$ (each prime factor is counted only once). For instance, if Alice erases the numbers $4$ and $6$, the prime factors of $ab = 2^3 \times 3$ and $2$ and $3$, and Alice writes $q = 6$ and $q^2 =36$. Prove that, after some time, and whatever Alice's strategy is, the list of numbers written on the blackboard will never change anymore. [i]Note: The order of the numbers of the list is not important.[/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].

Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$? [i]Proposed by Ray Li[/i]

Let $K$ be a cube with edge $n$, where $n>2$ is an even integer. Cube $K$ is divided into $n^3$ unit cubes. We call any set of $n^2$ unit cubes lying on the same horizontal or vertical level a layer. We dispose of $\frac{n^3}4$ colors, in each of which we paint exactly $4$ unit cubes. Prove that we can always select $n$ unit cubes of distinct colors, no two of which lie on the same layer.

Define the minimum real $C$ where for any reals $0 = a_0 < a_{1} < \dots < a_{1000}$ then $$\min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor.$

Evin’s calculator is broken and can only perform $3$ operations: Operation $1$: Given a number $x$, output $2x$. Operation $2$: Given a number $x$, output $4x +1$. Operation $3$: Given a number $x$, output $8x +3$. After initially given the number $0$, how many numbers at most $128$ can he make?

In the diagram $ABCDEFG$ is a regular heptagon (a $7$ sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\dfrac{m}{n}$ where m and n are relatively prime positive integers. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); string lab[]={"A","B","C","D","E","F","G"}; real r = 360/7; pair A=dir(90-r),B=dir(90),C=dir(90+r),D=dir(90+2*r),E=dir(90+3*r),F=dir(90+4*r),G=dir(90+5*r); draw(A--E--B--F--C--G--D--cycle); for(int k = -1;k <= 5;++k) { label("$"+lab[k+1]+"$",dir(90+k*r),dir(90+k*r)); } [/asy]

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

We are given a finite set of segments of the same line. Prove that we can color each segment red or blue such that, for each point $p$ on the line, the number of red segments containing $p$ differs from the number of blue segments containing $p$ by at most $1$.

Let $u_1$, $u_2$, $u_3$, $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$. Suppose $u_1 = 39$ and $u_2 = 45$. Prove that 1986 divides infinitely many terms of the sequence.

$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$ [asy] draw((0,0)--(10,0)--(3,-5)--(0,0)--(6.5,3)--(10,0)); draw((6.5,0)--(6.5,3)); draw((3,0)--(3,-5)); dot((0,0)); dot((10,0)); dot((3,0)); dot((3,-5)); dot((6.5,0)); dot((6.5,3)); label("A", (0,0), W); label("B", (6.5,3), N); label("C", (10,0), E); label("D", (3,-5), S); label("E", (3,0), N); label("F", (6.5,0), S);[/asy] $\text{(A)} \ 3.6 \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 4.2 \qquad \text{(D)} \ 4.5 \qquad \text{(E)} \ 5$

The lengths of the sides of a certain triangle and the diameter of the inscribed part circles are four consecutive terms of arithmetic progression. Find all such triangles.

Find all functions \(f:\mathbb{R}^{+} \to \mathbb{R}^{+}\) such that \[ f(x) > x \ \ \text{and} \ \ f(x-y+xy+f(y)) = f(x+y) + xf(y) \] for arbitrary positive real numbers \(x\) and \(y\).

In convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and \[ AB+DE = BC+EF = CD+FA. \] The midpoints of sides $AB$, $BC$, $DE$, $EF$ are $A_1$, $B_1$, $D_1$, $E_1$, and segments $A_1D_1$ and $B_1E_1$ meet at $O$. Prove that $\angle D_1OE_1 = \frac12 \angle DEF$.

Vlad draws 100 rays in the Euclidean plane. David then draws a line $\ell$ and pays Vlad one pound for each ray that $\ell$ intersects. Naturally, David wants to pay as little as possible. What is the largest amount of money that Vlad can get from David? [i]Proposed by Vlad Spătaru[/i]

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

A plane curve is parameterized by $x(t)=\displaystyle\int_{t}^{\infty} \dfrac {\cos u}{u} \, \mathrm{d}u $ and $ y(t) = \displaystyle\int_{t}^{\infty} \dfrac {\sin u}{u} \, \mathrm{d}u $ for $ 1 \le t \le 2 $. What is the length of the curve?

Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by \[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]

Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$

Let $n\ge 2$ be a given integer. Show that the number of strings of length $n$ consisting of $0'$s and $1'$s such that there are equal number of $00$ and $11$ blocks in each string is equal to \[2\binom{n-2}{\left \lfloor \frac{n-2}{2}\right \rfloor}\]

Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.

Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation: $|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.

Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the integers $x$ and $y$, where $x\leq y$ and replace them with $2x$ and $y-x.$

How many different solutions does the congruence $x^3+3x^2+x+3 \equiv 0 \pmod{25}$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6 $

For a finite group $G$ we denote by $n(G)$ the number of elements of the group and by $s(G)$ the number of subgroups of it. Decide whether the following statements are true or false. a) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}<a.$ b) For every $a>0$ the is a finite group $G$ with $\frac{n(G)}{s(G)}>a.$