A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$, and replace $X$ with $X+d$ if Neneng chose the sign "up" or $X-d$ if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero. Prove that if $n \geq 14$, Asep can win in at most $(n-5)/4$ steps.

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

There are $168$ primes below $1000$. Then sum of all primes below $1000$ is, $\textbf{(A)}~11555$ $\textbf{(B)}~76127$ $\textbf{(C)}~57298$ $\textbf{(D)}~81722$

Let $ABC$ be a triangle and $M$ the midpoint of $BC$. Parallels through $M$ to $AB$ and $AC$ intersect the tangent to $(ABC)$ at $A$ in $X$ and $Y$ respectively. Circles $(BMX)$ and $(CMY)$ intersect in $M$ and $S$. Prove that circles $(SXY)$ and $(SBC)$ are tangent. [i]Proposed by Ana Boiangiu[/i]

Let $x, y, z$ be real numbers such that $$\displaystyle{\begin{cases} x^2+y^2+z^2+(x+y+z)^2=9 \\ xyz \leq \frac{15}{32} \end{cases}} $$ Find the maximum possible value of $x.$

Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$?

Let $a_1, a_2, \ldots, a_{100}$ be positive integers, satisfying $$\frac{a_1^2+a_2^2+\ldots+a_{100}^2} {a_1+a_2+\ldots+a_{100}}=100.$$ What is the maximal value of $a_1$?

Bob writes a random string of $5$ letters, where each letter is either $A, B, C,$ or $D$. The letter in each position is independently chosen, and each of the letters $A, B, C, D$ is chosen with equal probability. Given that there are at least two $A's$ in the string, find the probability that there are at least three $A's$ in the string.

Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$. [i]Proposed by Marcin E. Kuczma, Poland[/i]

In his youth, Professor John Horton Conway lived on a farm with $2009$ cows. Conway wishes to move the cows from the negative $x$ axis to the positive $x$ axis. The cows are initially lined up in order $1, 2, . . . , 2009$ on the negative $x$ axis. Conway can give two possible commands to the cows. One is the PUSH command, upon which the first cow from the negative $x$ axis moves to the lowest position on the positive $y$ axis. The other is the POP command, upon which the cow in the lowest position on the $y$ axis moves to the positive $x$ axis. For example, if Conway says PUSH POP $2009$ times, then the resulting permutation of cows is the same, $1, 2, . . . , 2009$. If Conway says PUSH $2009$ times followed by POP $2009$ times, the resulting permutation of cows is $2009, . . . , 2, 1$. How many output permutations are possible after Conway finishes moving all the cows from the negative $x$ axis to the positive $x$ axis?

Let $x,y,z$ be distinct positive integers such that $(y+z)(z+x)=(x+y)^2$ . Show that \[x^2+y^2>8(x+y)+2(xy+1).\] (Paolo Leonetti)

The triangle $ABC$ is given. Consider the point $C'{}$ on the side $AB$ such that the segment $CC'$ divides the triangle into two triangles with equal radii of inscribed circles. Denote by $t_c$ the length of the segment $CC'$. Similarly, we define $t_a$ and $t_b$. Express the area of triangle $ABC$ in terms of $t_a,t_b$ and $t_c$. [i]Proposed by K. Mosevich[/i]

If $x_1,x_2,\ldots,x_n$ are arbitrary numbers from the interval $[0,2]$, prove that $$\sum_{i=1}^n\sum_{j=1}^n|x_i-x_j|\le n^2$$When is the equality attained?

Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

Ana choses two real numbers $ y>0,x $ and Bogdan repeatedly tries to guess these in the following manner: at step $ j $ he choses a real number $ b_j, $ asks her if $ b_j=x+jy, $ and she tells him the truth. [b]a)[/b] If $ x=0, $ can Bogdan find Ana's numbers in a finite number of steps? [b]b)[/b] If $ x\neq 0, $ can Bogdan find Ana's numbers in a finite number of steps?

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Today is Saturday, April $25$, $2020$. What is the value of $6 + 4 + 25 + 2020$? [b]p2.[/b] The figure below consists of a $2$ by $3$ grid of squares. How many squares of any size are in the grid? $\begin{tabular}{|l|l|l|} \hline & & \\ \hline & & \\ \hline \end{tabular}$ [b]p3.[/b] James is playing a game. He first rolls a six-sided dice which contains a different number on each side, then randomly picks one of twelve dierent colors, and finally ips a quarter. How many different possible combinations of a number, a color and a flip are there in this game? [b]p4.[/b] What is the sum of the number of diagonals and sides in a regular hexagon? [b]p5.[/b] Mickey Mouse and Minnie Mouse are best friends but they often fight. Each of their fights take up exactly one hour, and they always fight on prime days. For example, they fight on January $2$nd, $3$rd, but not the $4$th. Knowing this, how many total times do Mickey and Minnie fight in the months of April, May and June? [b]p6.[/b] Apple always loved eating watermelons. Normal watermelons have around $13$ black seeds and $25$ brown seeds, whereas strange watermelons had $45$ black seeds and $2$ brown seeds. If Apple bought $14$ normal watermelons and $7$ strange watermelons, then let $a$ be the total number of black seeds and $b$ be the total number of brown seeds. What is $a - b$? [b]p7.[/b] Jerry and Justin both roll a die once. The probability that Jerry's roll is greater than Justin's can be expressed as a fraction in the form $\frac{m}{n}$ in simplified terms. What is $m + n$? [b]p8.[/b] Taylor wants to color the sides of an octagon. What is the minimum number of colors Taylor will need so that no adjacent sides of the octagon will be filled in with the same color? [b]p9.[/b] The point $\frac23$ of the way from ($-6, 8$) to ($-3, 5$) can be expressed as an ordered pair $(a, b)$. What is $|a - b|$? [b]p10.[/b] Mary Price Maddox laughs $7$ times per class. If she teaches $4$ classes a day for the $5$ weekdays every week but doesn't laugh on Wednesdays, then how many times does she laugh after $5$ weeks of teaching? [b]p11.[/b] Let $ABCD$ be a unit square. If $E$ is the midpoint of $AB$ and $F$ lies inside $ABCD$ such that $CFD$ is an equilateral triangle, the positive difference between the area of $CED$ and $CFD$ can be expressed in the form $\frac{a-\sqrt{b}}{c}$ , where $a$, $b$, $c$ are in lowest simplified terms. What is $a + b + c$? [b]p12.[/b] Eddie has musician's syndrome. Whenever a song is a $C$, $A$, or $F$ minor, he begins to cry and his body becomes very stiff. On the other hand, if the song is in $G$ minor, $A$ at major, or $E$ at major, his eyes open wide and he feels like the happiest human being ever alive. There are a total of $24$ keys. How many different possibilities are there in which he cries while playing one song with two distinct keys? [b]p13.[/b] What positive integer must be added to both the numerator and denominator of $\frac{12}{40}$ to make a fraction that is equivalent to $\frac{4}{11}$ ? [b]p14.[/b] The number $0$ is written on the board. Each minute, Gene the genie either multiplies the number on the board by $3$ or $9$, each with equal probability, and then adds either $1$,$2$, or $3$, each with equal probability. Find the expected value of the number after $3$ minutes. [b]p15.[/b] $x$ satisfies $\dfrac{1}{x+ \dfrac{1}{1+\frac{1}{2}}}=\dfrac{1}{2+ \dfrac{1}{1- \dfrac{1}{2+\frac{1}{2}}}}$ Find $x$. [b]p16.[/b] How many different points in a coordinate plane can a bug end up on if the bug starts at the origin and moves one unit to the right, left, up or down every minute for $8$ minutes? [b]p17.[/b] The triplets Addie, Allie, and Annie, are racing against the triplets Bobby, Billy, and Bonnie in a relay race on a track that is $100$ feet long. The first person of each team must run around the entire track twice and tag the second person for the second person to start running. Then, the second person must run once around the entire track and tag the third person, and finally, the third person would only have to run around half the track. Addie and Bob run first, Allie and Billy second, Annie and Bonnie third. Addie, Allie, and Annie run at $50$ feet per minute (ft/m), $25$ ft/m, and $20$ ft/m, respectively. If Bob, Billy, and Bonnie run half as fast as Addie, Allie, and Annie, respectively, then how many minutes will it take Bob, Billy, and Bonnie to finish the race. Assume that everyone runs at a constant rate. [b]p18.[/b] James likes to play with Jane and Jason. If the probability that Jason and Jane play together is $\frac13$, while the probability that James and Jason is $\frac14$ and the probability that James and Jane play together is $\frac15$, then the probability that they all play together is $\frac{\sqrt{p}}{q}$ for positive integers $p$, $q$ where $p$ is not divisible by the square of any prime. Find $p + q$. [b]p19.[/b] Call an integer a near-prime if it is one more than a prime number. Find the sum of all near-primes less than$ 1000$ that are perfect powers. (Note: a perfect power is an integer of the form $n^k$ where $n, k \ge 2$ are integers.) [b]p20.[/b] What is the integer solution to $\sqrt{\frac{2x-6}{x-11}} = \frac{3x-7}{x+6}$ ? [b]p21.[/b] Consider rectangle $ABCD$ with $AB = 12$ and $BC = 4$ with $F$,$G$ trisecting $DC$ so that $F$ is closer to $D$. Then $E$ is on $AB$. We call the intersection of $EF$ and $DB$ $X$, and the intersection of $EG$ and $DB$ is $Y$. If the area of $\vartriangle XY E$ is \frac{8}{15} , then what is the length of $EB$? [b]p22.[/b] The sum $$\sum^{\infty}_{n=2} \frac{1}{4n^2-1}$$ can be expressed as a common fraction $\frac{a}{b}$ in lowest terms. Find $a + b$. [b]p23.[/b] In square $ABCD$, $M$, $N$, $O$, $P$ are points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, respectively. If $AB = 4$, $AM = BM$ and $DP = 3AP$, the least possible value of $MN + NO + OP$ can be expressed as $\sqrt{x}$ forsome integer x. Find x: [b]p24.[/b] Grand-Ovich the ant is at a vertex of a regular hexagon and he moves to one of the adjacent vertices every minute with equal probability. Let the probability that after $8$ minutes he will have returned to the starting vertex at least once be the common fraction $\frac{a}{b}$ in lowest terms. What is $a + b$? [b]p25.[/b] Find the last two non-zero digits at the end of $2020!$ written as a two digit number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$ (recalling that $ \sqrt2 = 1.414213..$.). (D. Fomin, Leningrad)

Given a polynomial $P(x)=x^4+3x^2-9x+1$. Calculate $P(\alpha^2+\alpha+1)$ where\[ \alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}} \]

Let $\gamma$ a circle and $P$ a point who lies outside the circle. Two arbitrary lines $l$ and $l'$ which pass through $P$ intersect the circle at the points $X$, $Y$ , respectively $X'$, $Y'$ , such that $X$ lies between $P$ and $Y$ and $X'$ lies between $P$ and $Y'$. Prove that the line determined by the circumcentres of the triangles $PXY'$ and $PX'Y$ passes through a fixed point.

Given that the line $y=mx+k$ intersects the parabola $y=ax^2+bx+c$ at two points, compute the product of the two $x$-coordinates of these points in terms of $a$, $b$, $c$, $k$, and $m$.

On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.

Find all natural numbers $n> 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$. (R. Henner, Vienna)

Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\] for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$? $\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$

The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$?

Prove that if a lattice triangle has no lattice points on its boundary in addition to its vertices, and one point in its interior, then this interior point is its center of gravity.