[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$, then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each. What is the least value of $k_1 + k_2 + k_3$ such that $B$ has a strategy to correctly guess the value of $n$ no matter what $A$ chooses?

Let $p_n$ denote the $n^{\text{th}}$ prime number and define $a_n=\lfloor p_n\nu\rfloor$ for all positive integers $n$ where $\nu$ is a positive irrational number. Is it possible that there exist only finitely many $k$ such that $\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all $i=1,2,\ldots,2020?$ [i]Proposed by Superguy and ayan.nmath[/i]

Let $ABC$ be an acute scalene triangle with incenter $I$, and let $M$ be the circumcenter of triangle $BIC$. Points $D$, $B'$, and $C'$ lie on side $BC$ so that $ \angle BIB' = \angle CIC' = \angle IDB = \angle IDC = 90^{\circ} $. Define $P = \overline{AB} \cap \overline{MC'}$, $Q = \overline{AC} \cap \overline{MB'}$, $S = \overline{MD} \cap \overline{PQ}$, and $K = \overline{SI} \cap \overline{DF}$, where segment $EF$ is a diameter of the incircle selected so that $S$ lies in the interior of segment $AE$. It is known that $KI=15x$, $SI=20x+15$, $BC=20x^{5/2}$, and $DI=20x^{3/2}$, where $x = \tfrac ab(n+\sqrt p)$ for some positive integers $a$, $b$, $n$, $p$, with $p$ prime and $\gcd(a,b)=1$. Compute $a+b+n+p$. [i]Proposed by Evan Chen[/i]

Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$

Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.

The twelve letters $A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$,$J$,$K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is $AB$, $CJ$, $DG$, $EK$, $FL$, $HI$. The probability that the last word listed contains $G$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ S$ be the set of all triangles $ ABC$ which have property: $ \tan A,\tan B,\tan C$ are positive integers. Prove that all triangles in $ S$ are similar.

For $k=1,2,\dots$, let $f_k$ be the number of times \[\sin\left(\frac{k\pi x}{2}\right)\] attains its maximum value on the interval $x\in[0,1]$. Compute \[\lim_{k\rightarrow\infty}\frac{f_k}{k}.\]

The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct.

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$. 1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

Consider a set of $1001$ points in the plane, no three collinear. Compute the minimum number of segments that must be drawn so that among any four points, we can find a triangle. [i]Proposed by Ahaan S. Rungta / Amir Hossein[/i]

We throw $ N$ balls into $ n$ urns, one by one, independently and uniformly. Let $ X_i\equal{}X_i(N,n)$ be the total number of balls in the $ i$th urn. Consider the random variable \[ y(N,n)\equal{}\min_{1 \leq i \leq n}|X_i\minus{}\frac Nn|.\] Verify the following three statements: (a) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow \infty$, then \[ P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}<x \right) \rightarrow 1\minus{}e^{\minus{}x\sqrt{2/ \pi}} \;\textrm{for all}\ \; x>0 \ .\] (b) If $ n\rightarrow \infty$ and $ N/n^3 \leq K$ ($ K$ constant), then for any $ \varepsilon > 0$ there is an $ A > 0$ such that \[ P(y(N,n) < A) > 1\minus{}\varepsilon .\] (c) If $ n \rightarrow \infty$ and $ N/n^3 \rightarrow 0$ then \[ P(y(N,n) < 1) \rightarrow 1.\] [i]P. Revesz[/i]

Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$ \[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]

Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$. [i]Holden Mui[/i]

Let $x, y, z, t$ be real numbers such that $(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2$. Prove that $x^2 + y^2 > 1$.

How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Let $ABCD$ be a cyclic quadrilateral inscirbed in a circle $O$ ($AB> AD$), and let $E$ be a point on segment $AB$ such that $AE = AD$. Let $AC \cap DE = F$, and $DE \cap O = K(\ne D)$. The tangent to the circle passing through $C,F,E$ at $E$ hits $AK$ at $L$. Prove that $AL = AD$ if and only if $\angle KCE = \angle ALE$.

Ria has $4$ green marbles and 8 red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is $\frac{p}{q}$ where the positive integers $p,q$ have no common factors other than $1$, what is $p+q$?

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Fifteen boys are standing on a field, and each of them has a ball. No two distances between two of the boys are equal. Each boy throws his ball to the boy standing closest to him. (a) Show that one of the boys does not get any ball. (b) Prove that none of the boys gets more than five balls.