Let $p(k)$ be the probability that if we choose a uniformly random subset $S$ of $\{1, 2, \ldots, 18\},$ then $|S| \equiv k \pmod{5}.$ Evaluate $$\sum_{k=0}^4 \left|p(k)-\frac{1}{5}\right|.$$

A [i]permutation[/i] of the integers $1901, 1902, \cdots, 2000$ is a sequence $a_1, a_2, \cdots, a_{100}$ in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums \[s_1 = a_1,\;\;s_2 = a_1 + a_2,\;\;s_3 = a_1 + a_2 + a_3, \; \ldots\;, \; s_{100} = a_1 + a_2 + \cdots + a_{100}.\] How many of these permutations will have no terms of the sequence $s_1, \ldots, s_{100}$ divisible by three?

Antonio and Bernardo play the following game. They are given two piles of chips, one with $m$ and the other with $n$ chips. Antonio starts, and thereafter they make in turn one of the following moves: (i) take a chip from one pile; (ii) take a chip from each of the piles; (ii) remove a chip from one of the piles and put it onto the other. Who cannot make any more moves, loses. Decide, as a function of $m$ and $n$ if one of the players has a winning strategy, and in the case of the affirmative answer describe that strategy.

Find all angles $0 < \theta < 90^\circ$ for which there exists an angle $0 < \beta < 90^\circ$ such that a right triangle with angles $90^\circ, \theta, 90^\circ - \theta$ can be tiled by a finite number of isosceles triangles with angles $\beta, \beta, 180^\circ - 2\beta$. (The isosceles triangles are not necessarily pairwise congruent, but they are pairwise similar.)

The figure shows $9$ circles connected by $12$ lines. Georg must colour each circle either red or blue. He gets one point for each line connecting circles with different colours. How many points can he at most achieve? [img]https://cdn.artofproblemsolving.com/attachments/3/9/983d3c5755547246899891db141fe2383f3dc1.png[/img]

We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), all different from $0$. What is the remainder of $k(n)$ on division by $8$?

Let $G$ be a connected simple graph. When we add an edge to $G$ (between two unconnected vertices), then using at most $17$ edges we can reach any vertex from any other vertex. Find the maximum number of edges to be used to reach any vertex from any other vertex in the original graph, i.e. in the graph before we add an edge.

Given a permutation ($a_0, a_1, \ldots, a_n$) of the sequence $0, 1,\ldots, n$. A transportation of $a_i$ with $a_j$ is called legal if $a_i=0$ for $i>0$, and $a_{i-1}+1=a_j$. The permutation ($a_0, a_1, \ldots, a_n$) is called regular if after a number of legal transportations it becomes ($1,2, \ldots, n,0$). For which numbers $n$ is the permutation ($1, n, n-1, \ldots, 3, 2, 0$) regular?

Arrange arbitrarily $1,2,\ldots ,25$ on a circumference. We consider all $25$ sums obtained by adding $5$ consecutive numbers. If the number of distinct residues of those sums modulo $5$ is $d$ $(0\le d\le 5)$,find all possible values of $d$.

Given a circle with its perimeter equal to $n$( $n \in N^*$), the least positive integer $P_n$ which satisfies the following condition is called the “[i]number of the partitioned circle[/i]”: there are $P_n$ points ($A_1,A_2, \ldots ,A_{P_n}$) on the circle; For any integer $m$ ($1\le m\le n-1$), there always exist two points $A_i,A_j$ ($1\le i,j\le P_n$), such that the length of arc $A_iA_j$ is equal to $m$. Furthermore, all arcs between every two adjacent points $A_i,A_{i+1}$ ($1\le i\le P_n$, $A_{p_n+1}=A_1$) form a sequence $T_n=(a_1,a_2,,,a_{p_n})$ called the “[i]sequence of the partitioned circle[/i]”. For example when $n=13$, the number of the partitioned circle $P_{13}$=4, the sequence of the partitioned circle $T_{13}=(1,3,2,7)$ or $(1,2,6,4)$. Determine the values of $P_{21}$ and $P_{31}$, and find a possible solution of $T_{21}$ and $T_{31}$ respectively.

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?

$10$ real numbers are given $a_1,a_2,\ldots ,a_{10} $, and the $45$ sums of two of these numbers are formed $a_i+a_j $, $1\leq i&lt;j\leq 10$ . It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the $45$ sums there are $k$ that are not integers and $45-k$ that are integers.

Several dwarves were lined up in a row, and then they lined up in a row in a different order. Is it possible that exactly one third of the dwarves have two new neighbours and exactly one third of the dwarves have only one new neighbour, if the number of the dwarves is a) 9; b) 12?

[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$? [b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$. [b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute. [b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$. [b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$. [b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number? Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$. [b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ : $$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$ Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$. [b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$. [b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$? [b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$ [b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$. [b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that $$x^2 + y^2 + xy = 9,$$ $$y^2 + z^2 + yz = 16,$$ $$x^2 + z^2 + xz = 25.$$ Find $xy + yz + xz$ (the answer is unique). [b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$. [b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$). [b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations: $$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Find all pairs $(a,b)$ of positive integers with $a > b$ and $a^2 -b^2 =111$. [b]p2.[/b] Alice drives at a constant rate of $2017$ miles per hour. Find all positive values of $x$ such that she can drive a distance of $x^2$ miles in a time of $x$ minutes. [b]p3.[/b] $ABC$ is a right triangle with right angle at $B$ and altitude $BH$ to hypotenuse $AC$. If $AB = 20$ and $BH = 12$, find the area of triangle $\vartriangle ABC$. [u]Round 2[/u] [b]p4.[/b] Regular polygons $P_1$ and $P_2$ have $n_1$ and $n_2$ sides and interior angles $x_1$ and $x_2$, respectively. If $\frac{n_1}{n_2}= \frac75$ and $\frac{x_1}{x_2}=\frac{15}{14}$ , find the ratio of the sum of the interior angles of $P_1$ to the sum of the interior angles of $P_2$. [b]p5.[/b] Joey starts out with a polynomial $f (x) = x^2 +x +1$. Every turn, he either adds or subtracts $1$ from $f$ . What is the probability that after $2017$ turns, $f$ has a real root? [b]p6.[/b] Find the difference between the greatest and least positive integer values $x$ such that $\sqrt[20]{\lfloor \sqrt[17]{x}\rfloor}=1$. [u]Round 3[/u] [b]p7.[/b] Let $ABCD$ be a square and suppose $P$ and $Q$ are points on sides $AB$ and $CD$ respectively such that $\frac{AP}{PB} = \frac{20}{17}$ and $\frac{CQ}{QD}=\frac{17}{20}$ . Suppose that $PQ = 1$. Find the area of square $ABCD$. [b]p8.[/b] If $$\frac{\sum_{n \ge 0} r^n}{\sum_{n \ge 0} r^{2n}}=\frac{1+r +r^2 +r^3 +...}{1+r^2 +r^4 +r^6 +...}=\frac{20}{17},$$ find $r$ . [b]p9.[/b] Let $\overline{abc}$ denote the $3$ digit number with digits $a,b$ and $c$. If $\overline{abc}_{10}$ is divisible by $9$, what is the probability that $\overline{abc}_{40}$ is divisible by $9$? [u]Round 4[/u] [b]p10.[/b] Find the number of factors of $20^{17}$ that are perfect cubes but not perfect squares. [b]p11.[/b] Find the sum of all positive integers $x \le 100$ such that $x^2$ leaves the same remainder as $x$ does upon division by $100$. [b]p12.[/b] Find all $b$ for which the base-$b$ representation of $217$ contains only ones and zeros. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $X$ be a finite set with $n\geqslant 3$ elements and let $A_1,A_2,\ldots, A_p$ be $3$-element subsets of $X$ satisfying $|A_i\cap A_j|\leqslant 1$ for all indices $i,j$. Show that there exists a subset $A{}$ of $X$ so that none of $A_1,A_2,\ldots, A_p$ is included in $A{}$ and $|A|\geqslant\lfloor\sqrt{2n}\rfloor$.

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8\}$. We want to color, using $k$ colors, all subsets of $3$ elements of $X$ in such a way that, two disjoint subsets have distinct colors. Prove that: (a) $4$ colors are sufficient; (b) $3$ colors are not sufficient.

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

$Abdulqodir$ cut out $2024$ congruent regular $n-$gons from a sheet of paper and placed these $n-$gons on the table such that some parts of each of these $n-$gons may be covered by others. We say that a vertex of one of the afore-mentioned $n-$gons is $visible$ if it is not in the interior of another $n-$gon that is placed on top of it. For any $n>2$ determine the minimum possible number of visible vertices. \\ Proposed by David Hrushka, Slovakia

Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) [hide]=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)[/hide]

Given a convex polygon with vertices at lattice points on a plane containing origin $O$. Let $V_1$ be the set of vectors going from $O$ to the vertices of the polygon, and $V_2$ be the set of vectors going from $O$ to the lattice points that lie inside or on the boundary of the polygon (thus, $V_1$ is contained in $V_2$.) Two grasshoppers jump on the whole plane: each jump of the first grasshopper shift its position by a vector from the set $V_1$, and the second by the set $V_2$. Prove that there exists positive integer $c$ that the following statement is true: if both grasshoppers can jump from $O$ to some point $A$ and the second grasshopper needs $n$ jumps to do it, then the first grasshopper can use at most $n+c$ jumps to do so.

The center of a circle and nine randomly selected points on this circle are colored in red. Every pair of those points is connected by a line segment, and every point of intersection of two line segments inside the circle is colored in red. What is the largest possible number of red points? A. $235$ B. $245$ C. $250$ D. $220$ E. $265$

Prove that for any positive integers $a$ and $b$ we have $$a+(-1)^b \sum^a_{m=0} (-1)^{\lfloor{\frac{bm}{a}\rfloor}} \equiv b+(-1)^a \sum^b_{n=0} (-1)^{\lfloor{\frac{an}{b}\rfloor}} \pmod{4}.$$