Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$ For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let $d(a) = \sum^{2003}_{k=1} \left(a_k - k \right)^2.$ [b]I.[/b] Find the least value of $d(a)$. Denote this least value by $d_0$. [b]II.[/b] Find all permutations $a \in A$ such that $d(a) = d_0$.

Suppose $n$ points are uniformly chosen at random on the circumference of the unit circle and that they are then connected with line segments to form an $n$-gon. What is the probability that the origin is contained in the interior of this $n$-gon? Give your answer in terms of $n$, and consider only $n \ge 3$.

[b]2.[/b] Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements.

p1. Let $U_n$ be a sequence of numbers that satisfy: $U_1=1$, $U_n=1+U_1U_2U_3...U_{n-1}$ for $n=2,3,...,2020$ Prove that $\frac{1}{U_1}+\frac{1}{U_2}+...+\frac{1}{U_{2019}}<2$ p2. If $a= \left \lceil \sqrt{2020+\sqrt{2020+...+\sqrt{2020}}} \right\rceil$ , $b= \left \lfloor \sqrt{1442+\sqrt{1442+...+\sqrt{1442}}} \right \rfloor$, and $c=a-b$, then determine the value of $c$. p3. Fajar will buy a pair of koi fish in the aquarium. If he randomly picks $2$ fish, then the probability that the $2$ fish are of the same sex is $1/2$. Prove that the number of koi fish in the aquarium is a perfect square. p4. A pharmacist wants to put $155$ ml of liquid into $3$ bottles. There are 3 bottle choices, namely a. Bottle A $\bullet$ Capacity: $5$ ml $\bullet$ The price of one bottle is $10,000$ Rp $\bullet$ If you buy the next bottle, you will get a $20\%$ discount, up to the $4$th purchase or if you buy $4$ bottles, get $ 1$ free bottle A b. Bottle B $\bullet$ Capacity: $8$ ml $\bullet$ The price of one bottle is $15.000$ Rp $\bullet$ If you buy $2$ : $20\%$ discount $\bullet$ If you buy $3$ : Free $ 1$ bottle of B c. Bottle C $\bullet$ Capacity : $14$ ml $\bullet$ Buy $ 1$ : $25.000$ Rp $\bullet$ Buy $2$ : Free $ 1$ bottle of A $\bullet$ Buy $3$ : Free $ 1$ bottle of B If in one purchase, you can only buy a maximum of $4$ bottles, then look for the possibility of pharmacists putting them in bottles so that the cost is minimal (bottles do not have to be filled to capacity). p5. Two circles, let's say $L_1$ and $L_2$ have the same center, namely at point $O$. Radius of $L_1$ is $10$ cm and radius of $L_2$ is $5$ cm. The points $A, B, C, D, E, F$ lie on $L_1$ so the arcs $AB,BC,CD,DE,EF,FA$ are equal. The points $P, Q, R$ lie on $L_2$ so that the arcs $PQ,QR,RS$ are equal and $PA=PF=QB=QC=RD=RD$ . Determine the area of ​​the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/b/5/0729eca97488ddfc82ab10eda02c708fecd7ae.png[/img]

For a nonzero integer $k$, denote by $\nu_2(k)$ the maximal nonnegative integer $t$ such that $2^t \mid k$. Given are $n (\ge 2)$ pairwise distinct integers $a_1, a_2, \ldots, a_n$. Show that there exists an integer $x$, distinct from $a_1, \ldots, a_n$, such that among $\nu_2(x - a_1), \ldots, \nu_2(x - a_n)$ there are at least $n/4$ odd numbers and at least $n/4$ even numbers.

[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice? [b]p2.[/b] There are $998$ candies and chocolate bars and $499$ bags. Each bag may contain two items (either two candies, or two chocolate bars, or one candy and one chocolate bar). Ann distributed candies and chocolate bars in such a way that half of the candies share a bag with a chocolate bar. Helen wants to redistribute items in the same bags in such a way that half of the chocolate bars would share a bag with a candy. Is it possible to achieve that? [b]p3.[/b] Insert in sequence $2222222222$ arithmetic operations and brackets to get the number $999$ (For instance, from the sequence $22222$ one can get the number $45$: $22*2+2/2 = 45$). [b]p4.[/b] Put numbers from $15$ to $23$ in a $ 3\times 3$ table in such a way to make all sums of numbers in two neighboring cells distinct (neighboring cells share one common side). [b]p5.[/b] All integers from $1$ to $200$ are colored in white and black colors. Integers $1$ and $200$ are black, $11$ and $20$ are white. Prove that there are two black and two white numbers whose sums are equal. [b]p6.[/b] Show that $38$ is the sum of few positive integers (not necessarily, distinct), the sum of whose reciprocals is equal to $1$. (For instance, $11=6+3+2$, $1/16+1/13+1/12=1$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Compute the greatest integer less than or equal to $$\frac{10 + 12 + 14 + 16 + 18 + 20}{21}$$ [b]p2.[/b] Let$ A = 1$.$B = 2$, $C = 3$, $...$, $Z = 26$. Find $A + B +M + C$. [b]p3.[/b] In Mr. M's farm, there are $10$ cows, $8$ chickens, and $4$ spiders. How many legs are there (including Mr. M's legs)? [b]p4.[/b] The area of an equilateral triangle with perimeter $18$ inches can be expressed in the form $a\sqrt{b}{c}$ , where $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime. Find $a + b + c$. [b]p5.[/b] Let $f$ be a linear function so $f(x) = ax + b$ for some $a$ and $b$. If $f(1) = 2017$ and $f(2) = 2018$, what is $f(2019)$? [b]p6.[/b] How many integers $m$ satisfy $4 < m^2 \le 216$? [b]p7.[/b] Allen and Michael Phelps compete at the Olympics for swimming. Allen swims $\frac98$ the distance Phelps swims, but Allen swims in $\frac59$ of Phelps's time. If Phelps swims at a rate of $3$ kilometers per hour, what is Allen's rate of swimming? The answer can be expressed as $m/n$ for relatively prime positive integers $m, n$. Find $m + n$. [b]p8.[/b] Let $X$ be the number of distinct arrangements of the letters in "POONAM," $Y$ be the number of distinct arrangements of the letters in "ALLEN" and $Z$ be the number of distinct arrangements of the letters in "NITHIN." Evaluate $\frac{X+Z}{Y}$ : [b]p9.[/b] Two overlapping circles, both of radius $9$ cm, have centers that are $9$ cm apart. The combined area of the two circles can be expressed as $\frac{a\pi+b\sqrt{c}+d}{e}$ where $c$ is not divisible by the square of any prime and the fraction is simplified. Find $a + b + c + d + e$. [b]p10.[/b] In the Boxborough-Acton Regional High School (BARHS), $99$ people take Korean, $55$ people take Maori, and $27$ people take Pig Latin. $4$ people take both Korean and Maori, $6$ people take both Korean and Pig Latin, and $5$ people take both Maori and Pig Latin. $1$ especially ambitious person takes all three languages, and and $100$ people do not take a language. If BARHS does not oer any other languages, how many students attend BARHS? [b]p11.[/b] Let $H$ be a regular hexagon of side length $2$. Let $M$ be the circumcircle of $H$ and $N$ be the inscribed circle of $H$. Let $m, n$ be the area of $M$ and $N$ respectively. The quantity $m - n$ is in the form $\pi a$, where $a$ is an integer. Find $a$. [b]p12.[/b] How many ordered quadruples of positive integers $(p, q, r, s)$ are there such that $p + q + r + s \le 12$? [b]p13.[/b] Let $K = 2^{\left(1+ \frac{1}{3^2} \right)\left(1+ \frac{1}{3^4} \right)\left(1+ \frac{1}{3^8}\right)\left(1+ \frac{1}{3^{16}} \right)...}$. What is $K^8$? [b]p14.[/b] Neetin, Neeton, Neethan, Neethine, and Neekhil are playing basketball. Neetin starts out with the ball. How many ways can they pass 5 times so that Neethan ends up with the ball? [b]p15.[/b] In an octahedron with side lengths $3$, inscribe a sphere. Then inscribe a second sphere tangent to the first sphere and to $4$ faces of the octahedron. The radius of the second sphere can be expressed in the form $\frac{\sqrt{a}-\sqrt{b}}{c}$ , where the square of any prime factor of $c$ does not evenly divide into $b$. Compute $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.

You are given positive integers $m, n>1$. Vasyl and Petryk play the following game: they take turns marking on the coordinate plane yet unmarked points of the form $(x, y)$, where $x, y$ are positive integers with $1 \leq x \leq m, 1 \leq y \leq n$. The player loses if after his move there are two marked points, the distance between which is not a positive integer. Who will win this game if Vasyl moves first and each player wants to win? [i]Proposed by Mykyta Kharin[/i]

Consider a board of $a \times b$, with $a$ and $b$ integers greater than or equal to $2$. Initially their squares are colored black and white like a chess board. The permitted operation consists of choosing two squares with a common side and recoloring them as follows: a white square becomes black; a black box turns green; a green box turns white. Determine for which values of $a$ and $b$ it is possible, by a succession of allowed operations, to make all the squares that were initially white end black and all the squares that were initially black end white. Clarification: Initially there are no green squares, but they appear after the first operation.

All positive integers are distributed among two disjoint sets $N_{1}$ and $N_{2}$ such that no difference of two numbers belonging to the same set is a prime greater than 100. Find all such distributions. [i]Proposed by N. Sedrakyan[/i]

Let $ n$ be a positive integer, and let $ \left( a_{1},\ a_{2} ,\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1},\ c_{2},\ ...,\ c_{n}\right)$ be three sequences of integers such that for any two distinct numbers $ i$ and $ j$ from the set $ \left\{ 1,2,...,n\right\}$, none of the seven integers $ a_{i}\minus{}a_{j}$; $ \left( b_{i}\plus{}c_{i}\right) \minus{}\left( b_{j}\plus{}c_{j}\right)$; $ b_{i}\minus{}b_{j}$; $ \left( c_{i}\plus{}a_{i}\right) \minus{}\left( c_{j}\plus{}a_{j}\right)$; $ c_{i}\minus{}c_{j}$; $ \left( a_{i}\plus{}b_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\right)$; $ \left( a_{i}\plus{}b_{i}\plus{}c_{i}\right) \minus{}\left( a_{j}\plus{}b_{j}\plus{}c_{j}\right)$ is divisible by $ n$. Prove that: [b]a)[/b] The number $ n$ is odd. [b]b)[/b] The number $ n$ is not divisible by $ 3$. [hide="Source of the problem"][i]Source of the problem:[/i] This question is a generalization of one direction of Theorem 2.1 in: Dean Alvis, Michael Kinyon, [i]Birkhoff's Theorem for Panstochastic Matrices[/i], American Mathematical Monthly, 1/2001 (Vol. 108), pp. 28-37. The original Theorem 2.1 is obtained if you require $ b_{i}\equal{}i$ and $ c_{i}\equal{}\minus{}i$ for all $ i$, and add in a converse stating that such sequences $ \left( a_{1},\ a_{2},\ ...,\ a_{n}\right)$, $ \left( b_{1},\ b_{2},\ ...,\ b_{n}\right)$ and $ \left( c_{1} ,\ c_{2},\ ...,\ c_{n}\right)$ indeed exist if $ n$ is odd and not divisible by $ 3$.[/hide]

Let $n>2$ be an integer$.$ We want to color in red exactly $n+1$ of the numbers $1,2,\cdots,2n-1, 2n$ so that there do not exists three distinct red integers $x,y,z$ satisfying $x+y=z.$ Prove that there is one and one only way to color the red numbers according to the given condition$.$

Geoff has an infinite stock of sweets, which come in $n$ flavours. He arbitrarily distributes some of the sweets amongst $n$ children (a child can get sweets of any subset of all flavours, including the empty set). Call a distribution $k-\textit{nice}$ if every group of $k$ children together has sweets in at least $k$ flavours. Find all subsets $S$ of $\{ 1, 2, \dots, n \}$ such that if a distribution of sweets is $s$-nice for all $s \in S$, then it is $s$-nice for all $s \in \{ 1, 2, \dots, n \}$. [i]Proposed by Kyle Hess, USA[/i]

Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?

A cake is prepared for a dinner party to which only $p$ or $q$ persons will come ($p$ and $q$ are given co-prime integers). Find the minimum number of pieces (not necessarily equal) into which the cake must be cut in advance so that the cake may be equally shared between the persons in either case. (D. Fomin, Leningrad)

Given a prime $p$ and a real number $\lambda \in (0,1)$. Let $s$ and $t$ be positive integers such that $s \leqslant t < \frac{\lambda p}{12}$. $S$ and $T$ are sets of $s$ and $t$ consecutive positive integers respectively, which satisfy $$\left| \left\{ (x,y) \in S \times T : kx \equiv y \pmod p \right\}\right| \geqslant 1 + \lambda s.$$Prove that there exists integers $a$ and $b$ that $1 \leqslant a \leqslant \frac{1}{ \lambda}$, $\left| b \right| \leqslant \frac{t}{\lambda s}$ and $ka \equiv b \pmod p$.

[b]p1.[/b] What is the area of a triangle with side lengths $ 6$, $ 8$, and $10$? [b]p2.[/b] Let $f(n) = \sqrt{n}$. If $f(f(f(n))) = 2$, compute $n$. [b]p3.[/b] Anton is buying AguaFina water bottles. Each bottle costs $14 $dollars, and Anton buys at least one water bottle. The number of dollars that Anton spends on AguaFina water bottles is a multiple of $10$. What is the least number of water bottles he can buy? [b]p4.[/b] Alex flips $3$ fair coins in a row. The probability that the first and last flips are the same can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p5.[/b] How many prime numbers $p$ satisfy the property that $p^2 - 1$ is not a multiple of $6$? [b]p6.[/b] In right triangle $\vartriangle ABC$ with $AB = 5$, $BC = 12$, and $CA = 13$, point $D$ lies on $\overline{CA}$ such that $AD = BD$. The length of $CD$ can then be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p7.[/b] Vivienne is deciding on what courses to take for Spring $2021$, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne have? [b]p8.[/b] Square $ABCD$ has side length $2$. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$. Compute the area of pentagon $AFECD$. [b]p9.[/b] At the Boba Math Tournament, the Blackberry Milk Team has answered $4$ out of the first $10$ questions on the Boba Round correctly. If they answer all $p$ remaining questions correctly, they will have answered exactly $\frac{9p}{5}\%$ of the questions correctly in total. How many questions are on the Boba Round? [b]p10.[/b] The sum of two positive integers is $2021$ less than their product. If one of them is a perfect square, compute the sum of the two numbers. [b]p11.[/b] Points $E$ and $F$ lie on edges $\overline{BC}$ and $\overline{DA}$ of unit square $ABCD$, respectively, such that $BE =\frac13$ and $DF =\frac13$ . Line segments $\overline{AE}$ and $\overline{BF}$ intersect at point $G$. The area of triangle $EFG$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] Compute the number of positive integers $n \le 2020$ for which $n^{k+1}$ is a factor of $(1+2+3+· · ·+n)^k$ for some positive integer $k$. [b]p13.[/b] How many permutations of $123456$ are divisible by their last digit? For instance, $123456$ is divisible by $6$, but $561234$ is not divisible by $4$. [b]p14.[/b] Compute the sum of all possible integer values for $n$ such that $n^2 - 2n - 120$ is a positive prime number. [b]p15. [/b]Triangle $\vartriangle ABC$ has $AB =\sqrt{10}$, $BC =\sqrt{17}$, and $CA =\sqrt{41}$. The area of $\vartriangle ABC$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p16.[/b] Let $f(x) = \frac{1 + x^3 + x^{10}}{1 + x^{10}}$ . Compute $f(-20) + f(-19) + f(-18) + ...+ f(20)$. [b]p17.[/b] Leanne and Jing Jing are walking around the $xy$-plane. In one step, Leanne can move from any point $(x, y)$ to $(x + 1, y)$ or $(x, y + 1)$ and Jing Jing can move from $(x, y)$ to $(x - 2, y + 5)$ or $(x + 3, y - 1)$. The number of ways that Leanne can move from $(0, 0)$ to $(20, 20)$ is equal to the number of ways that Jing Jing can move from $(0, 0)$ to $(a, b)$, where a and b are positive integers. Compute the minimum possible value of $a + b$. [b]p18.[/b] Compute the number positive integers $1 < k < 2021$ such that the equation $x +\sqrt{kx} = kx +\sqrt{x}$ has a positive rational solution for $x$. [b]p19.[/b] In triangle $\vartriangle ABC$, point $D$ lies on $\overline{BC}$ with $\overline{AD} \perp \overline{BC}$. If $BD = 3AD$, and the area of $\vartriangle ABC$ is $15$, then the minimum value of $AC^2$ is of the form $p\sqrt{q} - r$, where $p, q$, and $r$ are positive integers and $q$ is not divisible by the square of any prime number. Compute $p + q + r$. [b]p20. [/b]Suppose the decimal representation of $\frac{1}{n}$ is in the form $0.p_1p_2...p_j\overline{d_1d_2...d_k}$, where $p_1, ... , p_j$ , $d_1,... , d_k$ are decimal digits, and $j$ and $k$ are the smallest possible nonnegative integers (i.e. it’s possible for $j = 0$ or $k = 0$). We define the [i]preperiod [/i]of $\frac{1}{n}$ to be $j$ and the [i]period [/i]of $\frac{1}{n}$ to be $k$. For example, $\frac16 = 0.16666...$ has preperiod $1$ and period $1$, $\frac17 = 0.\overline{142857}$ has preperiod $0$ and period $6$, and $\frac14 = 0.25$ has preperiod $2$ and period $0$. What is the smallest positive integer $n$ such that the sum of the preperiod and period of $\frac{1}{n}$ is $ 8$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all positive integers $n$ for which the square $n \times n$ can be cut into squares $2\times 2$ and $3\times3$ (with the sides parallel to the sides of the big square).

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Every street in the city of Hamiltonville connects two squares, and every square may be reached by streets from every other. The governor discovered that if he closed all squares of any route not passing any square more than once, every remained square would be reachable from each other. Prove that there exists a circular route passing every square of the city exactly once. [i]Author: S. Berlov[/i]

A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n) \le 2^{mn}$. [asy] unitsize(0.4 cm); for(int i = 0; i <= 11; ++i) { draw((i,0)--(i,7)); } for(int j = 0; j <= 7; ++j) { draw((0,j)--(11,j)); } label("$\underbrace{\hspace{4.4 cm}}$", (11/2,-0.5)); label("$\left. \begin{array}{c} \vspace{2.4 cm} \end{array} \right\}$", (11,7/2)); label("$m$ blocks", (11/2,-1.5)); label("$n$ blocks", (14,7/2)); [/asy]

The flights map of air company $K_{r,r}$ presents several cities. Some cities are connected by a direct (two way) flight, the total number of flights is m. One must choose two non-intersecting groups of r cities each so that every city of the first group is connected by a flight with every city of the second group. Prove that number of possible choices does not exceed $2*m^r$ .

A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer. (A): $30$ (B): $35$ (C): $40$ (D): $45$ (E): None of the above.

At the junction of some countably infinite number of roads sits a greyhound. On one of the roads a hare runs, away from the junction. The only thing known is that the (maximal) speed of the hare is strictly less than the (maximal) speed of the greyhound (but not their precise ratio). Does the greyhound have a strategy for catching the hare in a finite amount of time? ([i]Dan Schwarz[/i])