Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies \[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]

[u]Round 1[/u] [b]p1.[/b] Solve the maze [img]https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png[/img] [b]p2.[/b] Billiam can write a problem in $30$ minutes, Jerry can write a problem in $10$ minutes, and Evin can write a problem in $20$ minutes. Billiam begins writing problems alone at $3:00$ PM until Jerry joins himat $4:00$ PM, and Evin joins both of them at $4:30$ PM. Given that they write problems until the end of math team at $5:00$ PM, how many full problems have they written in total? [b]p3.[/b] How many pairs of positive integers $(n,k)$ are there such that ${n \choose k}= 6$? [u]Round 2 [/u] [b]p4.[/b] Find the sumof all integers $b > 1$ such that the expression of $143$ in base $b$ has an even number of digits and all digits are the same. [b]p5.[/b] Ιni thinks that $a \# b = a^2 - b$ and $a \& b = b^2 - a$, while Mimi thinks that $a \# b = b^2 - a$ and $a \& b = a^2 - b$. Both Ini and Mimi try to evaluate $6 \& (3 \# 4)$, each using what they think the operations $\&$ and $\#$ mean. What is the positive difference between their answers? [b]p6.[/b] A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count. [u]Round 3[/u] [b]p7.[/b] Maya wants to buy lots of burgers. A burger without toppings costs $\$4$, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy $1$ burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter. [b]p8.[/b] Consider square $ABCD$ and right triangle $PQR$ in the plane. Given that both shapes have area $1$, $PQ =QR$, $PA = RB$, and $P$, $A$, $B$ and $R$ are collinear, find the area of the region inside both square $ABCD$ and $\vartriangle PQR$, given that it is not $0$. [b]p9.[/b] Find the sum of all $n$ such that $n$ is a $3$-digit perfect square that has the same tens digit as $\sqrt{n}$, but that has a different ones digit than $\sqrt{n}$. [u]Round 4[/u] [b]p10.[/b] Jeremy writes the string: $$LMTLMTLMTLMTLMTLMT$$ on a whiteboard (“$LMT$” written $6$ times). Find the number of ways to underline $3$ letters such that from left to right the underlined letters spell LMT. [b]p11.[/b] Compute the remainder when $12^{2022}$ is divided by $1331$. [b]p12.[/b] What is the greatest integer that cannot be expressed as the sum of $5$s, $23$s, and $29$s? [u]Round 5 [/u] [b]p13.[/b] Square $ABCD$ has point $E$ on side $BC$, and point $F$ on side $CD$, such that $\angle EAF = 45^o$. Let $BE = 3$, and $DF = 4$. Find the length of $FE$. [b]p14.[/b] Find the sum of all positive integers $k$ such that $\bullet$ $k$ is the power of some prime. $\bullet$ $k$ can be written as $5654_b$ for some $b > 6$. [b]p15.[/b] If $\sqrt[3]{x} + \sqrt[3]{y} = 2$ and $x + y = 20$, compute $\max \,(x, y)$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence $(a_1,a_2,...,a_{675})$ is given so that each term is an alphabet in the English language (no distinction is made between lower and upper case letters). It is known that in the sequence $a$ is never followed by $b$ and $c$ is never followed by $d$. Show that there are integers $m$ and $n$ with $1 \le m < n \le 674$ such that $a_m = a_n$ and $a_{m+1} = a_{n+1}$·

Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds: \[\sum_{P\in\mathcal{C}\cap M} f(P)=0\] [i]Dorel Mihet[/i]

[b]p1.[/b] An evil galactic empire is attacking the planet Naboo with numerous automatic drones. The fleet defending the planet consists of $101$ ships. By the decision of the commander of the fleet, some of these ships will be used as destroyers equipped with one rocket each or as rocket carriers that will supply destroyers with rockets. Destroyers can shoot rockets so that every rocket destroys one drone. During the attack each carrier will have enough time to provide each destroyer with one rocket but not more. How many destroyers and how many carriers should the commander assign to destroy the maximal number of drones and what is the maximal number of drones that the fleet can destroy? [b]p2.[/b] Solve the inequality: $\sqrt{x^2-3x+2} \le \sqrt{x+7}$ [b]p3.[/b] Find all positive real numbers $x$ and $y$ that satisfy the following system of equations: $$x^y = y^{x-y}$$ $$x^x = y^{12y}$$ [b]p4.[/b] A convex quadrilateral $ABCD$ with sides $AB = 2$, $BC = 8$, $CD = 6$, and $DA = 7$ is divided by a diagonal $AC$ into two triangles. A circle is inscribed in each of the obtained two triangles. These circles touch the diagonal at points $E$ and $F$. Find the distance between the points $E$ and $F$. [b]p5.[/b] Find all positive integer solutions $n$ and $k$ of the following equation: $$\underbrace{11... 1}_{n} \underbrace{00... 0}_{2n+3} + \underbrace{77...7}_{n+1} \underbrace{00...0}_{n+1}+\underbrace{11...1}_{n+2} = 3k^3.$$ [b]p6.[/b] The Royal Council of the planet Naboo consists of $12$ members. Some of these members mutually dislike each other. However, each member of the Council dislikes less than half of the members. The Council holds meetings around the round table. Queen Amidala knows about the relationship between the members so she tries to arrange their seats so that the members that dislike each other are not seated next to each other. But she does not know whether it is possible. Can you help the Queen in arranging the seats? Justify your answer. PS. You should use hide for answers.

[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$? [b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails? [b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$. [b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$? [b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average. [b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number. [b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle. [b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping. [b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$. [b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] How many ways can we arrange the elements $\{1, 2, ..., n\}$ to a sequence $a_1, a_2, ..., a_n$ such that there is only exactly one $a_i$, $a_{i+1}$ such that $a_i > a_{i+1}$? [b]p2. [/b]How many distinct (non-congruent) triangles are there with integer side-lengths and perimeter $2012$? [b]p3.[/b] Let $\phi$ be the Euler totient function, and let $S = \{x| \frac{x}{\phi (x)} = 3\}$. What is $\sum_{x\in S} \frac{1}{x}$? [b]p4.[/b] Denote $f(N)$ as the largest odd divisor of $N$. Compute $f(1) + f(2) + f(3) +... + f(29) + f(30)$. [b]p5.[/b] Triangle $ABC$ has base $AC$ equal to $218$ and altitude $100$. Squares $s_1, s_2, s_3, ...$ are drawn such that $s_1$ has a side on $AC$ and has one point each touching $AB$ and $BC$, and square $s_k$ has a side on square $s_{k-}1$ and also touches $AB$ and $BC$ exactly once each. What is the sum of the area of these squares? [b]p6.[/b] Let $P$ be a parabola $6x^2 - 28x + 10$, and $F$ be the focus. A line $\ell$ passes through $F$ and intersects the parabola twice at points $P_1 = (2,-22)$, $P_2$. Tangents to the parabola with points at $P_1, P_2$ are then drawn, and intersect at a point $Q$. What is $m\angle P_1QP_2$? PS. You had better use hide for answers.

Let's call a checkered polygon a [i]strip[/i], which can be traversed entirely, starting from some of its cells and then moving only in two directions - up or to the right. Several such strips can be inserted into each other by shifting by a vector $(-1.1)$. Prove that for any strip consisting of an even number of cells, there is such an odd $k$ that if you combine $k$ of the same strips by inserting them sequentially into each other, then the resulting polygon can be divided along the grid lines into two equal parts. [i]Proposed by I. Markelov, S. Markelov[/i]

It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that $i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements $ii)$ Upper bound can be reached for $n=3$

Bethan is playing a game on an $n\times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves Bethan realises she cannot make any more moves. Determine the minimum value of $M$. [i]United Kingdom, Sam Bealing[/i]

Let $a_1, a_2, ..., a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \le i < j \le n$, there exist $k,\ell$, both different from $i$ and $j$ such that $a_i + a_j = a_k + a_{\ell}$. What is the smallest possible value of $n$?

A set of balls contains $ n$ balls which are labeled with numbers $ 1,2,3,\ldots,n.$ We are given $ k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that (a) the balls labeled with the same number are of the same colour, (b) any subset of $ k\plus{}1$ balls with (not necessarily different) labels $ a_{1},a_{2},\ldots,a_{k\plus{}1}$ satisfying the condition $ a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}$, contains at least one ball of each colour. Find, depending on $ k$ the greatest possible number $ n$ which admits such a colouring.

A prefrosh is participating in Caltech’s “Rotation.” They must rank Caltech’s $8$ houses, which are Avery, Page, Lloyd, Venerable, Ricketts, Blacker, Dabney, and Fleming, each a distinct integer rating from $1$ to $8$ inclusive. The conditions are that the rating $x$ they give to Fleming is at most the average rating $y$ given to Ricketts, Blacker, and Dabney, which is in turn at most the average rating $z$ given to Avery, Page, Lloyd, and Venerable. Moreover $x, y, z$ are all integers. How many such rankings can the prefrosh provide?

There are $1978$ points in the flat plane. Each point has a circular disk with that point as its center and the radius is the distance to a fixed point. Prove that there are five of these circular disks, which together cover all $1978$ points (circular disk means: the circle and its inner area).

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

In the Euclidean plane, let be a point $ O $ and a finite set $ \mathcal{M} $ of points having at least two points. Prove that there exists a proper subset of $ \mathcal{M}, $ namely $ \mathcal{M}_0, $ such that the following inequality is true: $$ \sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ $$

Does there exist a set $S$ of $100$ points in a plane such that the center of mass of any $10$ points in $S$ is also a point in $S$?

[hide=Rules]Time Limit: 25 minutes Maximum Possible Score: 81 The following is a mathematical Sudoku puzzle which is also a crossword. Your job is to fill in as many blanks as you possibly can, including all shaded squares. You do not earn extra points for showing your work; the only points you get are for correctly filled-in squares. You get one point for each correctly filled-in square. You should read through the following rules carefully before starting. $\bullet$ Your time limit for this round is $25$ minutes, in addition to the five minutes you get for reading the rules. So make use of your time wisely. The round is based more on speed than on perfect reasoning, so use your intuition well, and be fast. $\bullet$ This is a Sudoku puzzle; all the squares should be filled in with the digits $1$ through $9$ so that every row and column contains each digit exactly once. In addition, each of the nine $3\times 3$ boxes that compose the grid also contains each digit exactly once. Furthermore, this is a super-Sudoku puzzle; in addition to satisfying all these conditions, the four $3\times 3$ boxes with red outlines also contain each of $1,..., 9$ exactly once. This last property is important to keep in mind – it may help you solve the puzzle faster. $\bullet$ Just to restate the idea, you can use the digits $1$ through $9$, but not $0$. You may not use any other symbol, such as $\pi$ or $e$ or $\epsilon$. Each square gets exactly one digit. $\bullet$ The grid is also a crossword puzzle; the usual rules apply. The shaded grey squares are the “black” squares of an ordinary crossword puzzle. The white squares as well as the shaded yellow ones count as the “white” crossword squares. All squares, white or shaded, count as ordinary Sudoku squares. $\bullet$ If you obtain the unique solution to the crossword puzzle, then this solution extends to a unique solution to the Sudoku puzzle. $\bullet$ You may use a graphing calculator to help you solve the clues. The following hints and tips may prove useful while solving the puzzle. $\bullet$ Use the super-Sudoku structure described in the first rule; use all the symmetries you have. Remember that we are not looking for proofs or methods, only for correctly filled-in squares. $\bullet$ If you find yourself stuck on a specific clue, it is nothing to worry about. You can obtain the solution to that clue later on by solving other clues and figuring out certain digits of your desired solution. Just move on to the rest of the puzzle. $\bullet$ As you progress through the puzzle, keep filling in all squares you have found on your solution sheet, including the shaded ones. Remember that for scoring, the shaded grey squares count the same as the white ones. Good luck! [/hide] [asy] // place label "s" in row i, column j void labelsq(int i, int j, string s) { label("$"+s+"$",(j-0.5,7.5-i),fontsize(14)); } // for example, use the command // labelsq(1,7,"2"); // to put the digit 2 in the top right box // **** rest of code **** size(250); defaultpen(linewidth(1)); pair[] labels = {(1,1),(1,4),(1,6),(1,7),(1,9),(2,1),(2,6),(3,4),(4,1),(4,8),(5,1),(6,3),(6,5),(6,6),(7,1),(7,2),(7,7),(7,9),(8,1),(8,4),(9,1),(9,6)}; pair[] blacksq = {(1,5),(2,5),(3,2),(3,3),(3,8),(5,5),(5,6),(5,7),(5,9),(6,2),(6,7),(6,9),(8,3),(9,5),(9,8)}; path peachsq = shift(1,1)*scale(3)*unitsquare; pen peach = rgb(0.98,0.92,0.71); pen darkred = red + linewidth(2); fill(peachsq,peach); fill(shift(4,0)*peachsq,peach); fill(shift(4,4)*peachsq,peach); fill(shift(0,4)*peachsq,peach); for(int i = 0; i < blacksq.length; ++i) fill(shift(blacksq[i].y-1, 9-blacksq[i].x)*unitsquare, gray(0.6)); for(int i = 0; i < 10; ++i) { pen sudokuline = linewidth(1); if(i == 3 || i == 6) sudokuline = linewidth(2); draw((0,i)--(9,i),sudokuline); draw((i,0)--(i,9),sudokuline); } draw(peachsq,darkred); draw(shift(4,0)*peachsq,darkred); draw(shift(4,4)*peachsq,darkred); draw(shift(0,4)*peachsq,darkred); for(int i = 0; i < labels.length; ++i) label(string(i+1), (labels[i].y-1, 10-labels[i].x), SE, fontsize(10)); // **** draw letters **** draw(shift(.5,.5)*((0,6)--(0,8)--(2,8)--(2,7)--(0,7)^^(3,8)--(3,6)--(5,6)--(5,8)^^(6,6)--(6,8)--(7,8)--(7,7)--(7,8)--(8,8)--(8,6)^^(0,3)--(0,5)--(2,5)--(2,3)--(2,4)--(0,4)^^(5,3)--(3,3)--(3,5)--(5,5)),linewidth(1)+rgb(0.94,0.74,0.58)); // **** end rest of code ****[/asy] [b][u][i]Across[/i][/u][/b] [b]1 Across.[/b] The following is a normal addition where each letter represents a (distinct) digit: $$GOT + TO + GO + TO = TOP$$This certainly does not have a unique solution. However, you discover suddenly that $G = 2$ and $P \notin \{4, 7\}$. Then what is the numeric value of the expression $GOT \times TO$? [b]3 Across.[/b] A strobogrammatic number which reads the same upside down, e.g. $619$. On the other hand, a triangular number is a number of the form $n(n + 1)/2$ for some $n \in N$, e.g. $15$ (therefore, the $i^{th}$ triangular number $T_i$ is the sum of $1$ through $i$). Let $a$ be the third strobogrammatic prime number. Let $b$ be the smaller number of the smallest pair of triangular numbers whose sum and difference are also triangular numbers. What is the value of $ab$? [b]6 Across.[/b] A positive integer $m$ is said to be palindromic in base $\ell$ if, when written in base $\ell$ , its digits are the same front-to-back and back-to-front. For $j, k \in N$, let $\mu (j, k)$ be the smallest base-$10$ integer that is palindromic in base $j$ as well as in base$ k$, and let $\nu (j, k) := (j + k) \cdot \mu (j, k)$. Find the value of $\nu (5, 9)$. [b]7 Across.[/b] Suppose you have the unique solution to this Sudoku puzzle. In that solution, let $X$ denote the sum of all digits in the shaded grey squares. Similarly, let $Y$ denote the sum of all numbers in the shaded yellow squares on the upper left block (i.e. the $3 \times 3$ box outlined red towards the top left). Concatenate $X$ with $Y$ in that order, and write that down. [b]8 Across.[/b] For any $n \in N$ such that $1 < n < 10$, define the sequence $X_{n,1}$,$X_{n,2}$,$ ...$ by $X_{n,1} = n$, and for $r \ge 2$, X_{n,r} is smallest number $k \in N$ larger than X_{n,r-1} such that $k$ and the sum of digits of $k$ are both powers of $n$. For instance, $X_{3,1 = 3}$, $X_{3,2} = 9$, $X_{3,3} = 27$, and so on. Concatenate $X_{2,2}$ with $X_{2,4}$, and write down the answer. [b]9 Across.[/b] Find positive integers $x, y,z$ satisfying the following properties: $y$ is obtained by subtracting $93$ from $x$, and $z$ is obtained by subtracting $183$ from $y$, furthermore, $x, y$ and $z$ in their base-$10$ representations contain precisely all the digits from $1$ through $9$ once (i.e. concatenating $x, y$ and $z$ yields a valid $9$-digit Sudoku answer). Obviously, write down the concatenation of $x, y$ and $z$ in that order. [b]11 Across.[/b] Find the largest pair of two-digit consecutive prime numbers $a$ and $b$ (with $a < b$) such that the sum of the digits of a plus the sum of the digits of b is also a prime number. Write the concatenation of $a$ and $b$. [b]12 Across.[/b] Suppose you have a strip of $2n + 1$ squares, with n frogs on the $n$ squares on the left, and $n$ toads on the $n$ squares on the right. A move consists either of a toad or a frog sliding to an adjacent square if it is vacant, or of a toad or a frog jumping one square over another one and landing on the next square if it is vacant. For instance, the starting position [img]https://cdn.artofproblemsolving.com/attachments/a/a/6c97f15304449284dc282ff86014f526322e4a.png[/img] has the position [img]https://cdn.artofproblemsolving.com/attachments/e/6/e2c9520731bd94dc0aa37f540c2b9d1bce6432.png[/img] or the position [img]https://cdn.artofproblemsolving.com/attachments/3/f/06868eca80d649c4f80425dc9dc5c596cb2a4e.png[/img] as results of valid first moves. What is the minimum number of moves needed to swap the toads with the frogs if $n = 5$? How about $n = 6$? Concatenate your answers. [b]15 Across.[/b] Let $w$ be the largest number such that $w$, $2w$ and $3w$ together contain every digit from $1$ through $9$ exactly once. Let $x$ be the smallest integer with the property that its first $5$ multiples contain the digit $9$. A Leyland number is an integer of the form $m^n + n^m$ for integers $m, n > 1$. Let $y$ be the fourth Leyland number. A Pillai prime is a prime number $p$ for which there is an integer $n > 0$ such that $n! \equiv - 1 (mod \,\, p)$, but $p \not\equiv 1 (mod \,\, n)$. Let $z$ be the fourth Pillai prime. Concatenate $w$, $x, y$ and $z$ in that order to obtain a permutation of $1,..., 9$. Write down this permutation. [b]19 Across.[/b] A hoax number $k \in N$ is one for which the sum of its digits (in base $10$) equals the sum of the digits of its distinct prime factors (in base $10$). For instance, the distinct prime factors of $22$ are $2$ and $11$, and we have $2+2 = 2+(1+1)$. In fact, $22$ is the first hoax number. What is the second? [b]20 Across.[/b] Let $a, b$ and $c$ be distinct $2$-digit numbers satisfying the following properties: – $a$ is the largest integer expressible as $a = x^y = y^x$, for distinct integers $x$ and $y$. – $b$ is the smallest integer which has three partitions into three parts, which all give the same product (which turns out to be $1200$) when multiplied. – $c$ is the largest number that is the sum of the digits of its cube. Concatenate $a, b$ and $c$, and write down the resulting 6-digit prime number. [b]21 Across.[/b] Suppose $N = \underline{a}\, \underline{b} \, \underline{c} \, \underline{d}$ is a $4$-digit number with digits $a, b, c$ and $d$, such that $N = a \cdot b \cdot c \cdot d^7$. Find $N$. [b]22 Across.[/b] What is the smallest number expressible as the sum of $2, 3, 4$, or $5$ distinct primes? [b][u][i]Down [/i][/u][/b] [b]1 Down.[/b] For some $a, b, c \in N$, let the polynomial $$p(x) = x^5 - 252x^4 + ax^3 - bx^2 + cx - 62604360$$ have five distinct roots that are positive integers. Four of these are 2-digit numbers, while the last one is single-digit. Concatenate all five roots in decreasing order, and write down the result. [b]2 Down.[/b] Gene, Ashwath and Cosmin together have $2511$ math books. Gene now buys as many math books as he already has, and Cosmin sells off half his math books. This leaves them with $2919$ books in total. After this, Ashwath suddenly sells off all his books to buy a private jet, leaving Gene and Cosmin with a total of $2184$ books. How many books did Gene, Ashwath and Cosmin have to begin with? Concatenate the three answers (in the order Gene, Ashwath, Cosmin) and write down the result. [b]3 Down.[/b] A regular octahedron is a convex polyhedron composed of eight congruent faces, each of which is an equilateral triangle; four of them meet at each vertex. For instance, the following diagram depicts a regular octahedron: [img]https://cdn.artofproblemsolving.com/attachments/c/1/6a92f12d5e9f56b0699531ae8369a0ab8ab813.png[/img] Let $T$ be a regular octahedron of edge length $28$. What is the total surface area of $T$ , rounded to the nearest integer? [b]4 Down.[/b] Evaluate the value of the expression $$\sum^{T_{25}}_{k=T_{24}+1}k, $$ where $T_i$ denotes the $i^{th}$ triangular number (the sum of the integers from $1$ through $i$). [b]5 Down.[/b] Suppose $r$ and $s$ are consecutive multiples of$ 9$ satisfying the following properties: – $r$ is the smallest positive integer that can be written as the sum of $3$ positive squares in $3$ different ways. – $s$ is the smallest $2$-digit number that is a Woodall number as well as a base-$10$ Harshad number. A Woodall number is any number of the form $n \cdot 2^n - 1$ for some $n \in N$. A base-$10$ Harshad number is divisible by the sum of its digits in base $10$. Concatenate $r$ and $s$ and write down the result. [b]10 Down.[/b] For any $k \in N$, let $\phi_p(k)$ denote the sum of the distinct prime factors of $k$. Suppose $N$ is the largest integer less than $50000$ satisfying $\phi_p(N) =\phi_p(N + 1)$, where the common value turns out to be a meager $55$. What is$ N$? [b]13 Down.[/b] The $n^{th}$ $s$-gonal number $P(s, n)$ is defined as $$P(s, n) = (s - 3)T_{n-1} + T_n$$ where $T_i$ is the $i^{th}$ triangular number (recall that the $i^{th}$ triangular number is the sum of the numbers $1$ through $i$). Find the least $N$ such that $N$ is both a $34$-gonal number, and a $163$-gonal number. [b]14 Down.[/b] A biprime is a positive integer that is the product of precisely two (not necessarily distinct) primes. A cluster of biprimes is an ordered triple $(m,m + 1,m + 2)$ of consecutive integers that are biprimes. There are precisely three clusters of biprimes below 100. Denote these by, say, $$\{(p, p + 1, p + 2), (q, q + 1,q + 2), (r, r + 1, r + 2)\}$$ and add the condition that $p + 2 < q < r - 2$ to fix the three clusters. Interestingly, $p + 1$ and $q$ are both multiples of $17$. Concatenate $q$ with $p + 1$ in that order, and write down the result. [b]16 Down.[/b] Find the least positive integer $m$ (written in base $10$ as $m = \underline{a} \, \underline{b} \, \underline{c} $, with digits $a, b,c$), such that $m = (b + c)^a$. [b]17 Down.[/b] Let $X$ be a set containing $32$ elements, and let $Y\subseteq X$ be a subset containing $29$ elements. How many $2$-element subsets of $X$ are there which have nonempty intersection with $Y$? [b]18 Down.[/b] Find a positive integer $K < 196$, which is a strange twin of the number $196$, in the sense that $K^2$ shares the same digits as $196^2$, and $K^3$ shares the same digits as $196^3$. PS. You should use hide for answers.

A square \( K = 2025 \times 2025 \) is given. We define a [i]stick[/i] as a rectangle where one of its sides is \( 1 \), and the other side is a positive integer from \( 1 \) to \( 2025 \). Find the largest positive integer \( C \) such that the following condition holds: [list] [*] If several sticks with a total area not exceeding \( C \) are taken, it is always possible to place them inside the square \( K \) so that each stick fully completely covers an integer number of \( 1 \times 1 \) squares, and no \( 1 \times 1 \) square is covered by more than one stick. [/list] [i](Basically, you can rotate sticks, but they have to be aligned by lines of the grid)[/i] [i]Proposed by Anton Trygub[/i]

$200$ soldiers occupy in a rectangle (military call it a square and educated military a carree): $20$ men (per row) times $10$ men (per column). In each row, we consider the tallest man (if some are of equal height, choose any of them) and of the $10$ men considered we select the shortest (if some are of equal height, choose any of them). Call him $A$. Next the soldiers assume their initial positions and in each column the shortest soldier is selected, of these $20$, the tallest is chosen. Call him $B$. Two colonels bet on which of the two soldiers chosen by these two distinct procedures is taller: $A$ or $B$. Which colonel wins the bet?

Let $ f(n,k)$ be the number of ways of distributing $ k$ candies to $ n$ children so that each child receives at most $ 2$ candies. For example $ f(3,7) \equal{} 0,f(3,6) \equal{} 1,f(3,4) \equal{} 6$. Determine the value of $ f(2006,1) \plus{} f(2006,4) \plus{} \ldots \plus{} f(2006,1000) \plus{} f(2006,1003) \plus{} \ldots \plus{} f(2006,4012)$.

Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.

Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C, and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be followed by an upper-case C, and no yellow C can be followed by a green C. In how many ways can the Cs be written?

[b]p1.[/b] Jose,, Luciano, and Placido enjoy playing cards after their performances, and you are invited to deal. They use just nine cards, numbered from $2$ through $10$, and each player is to receive three cards. You hope to hand out the cards so that the following three conditions hold: A) When Jose and Luciano pick cards randomly from their piles, Luciano most often picks a card higher than Jose, B) When Luciano and Placido pick cards randomly from their piles, Placido most often picks a card higher than Luciano, C) When Placido and Jose pick cards randomly from their piles, Jose most often picks a card higher than Placido. Explain why it is impossible to distribute the nine cards so as to satisfy these three conditions, or give an example of one such distribution. [b]p2.[/b] Is it possible to fill a rectangular box with a finite number of solid cubes (two or more), each with a different edge length? Justify your answer. [b]p3.[/b] Two parallel lines pass through the points $(0, 1)$ and $(-1, 0)$. Two other lines are drawn through $(1, 0)$ and $(0, 0)$, each perpendicular to the ¯rst two. The two sets of lines intersect in four points that are the vertices of a square. Find all possible equations for the first two lines. [b]p4.[/b] Suppose $a_1, a_2, a_3,...$ is a sequence of integers that represent data to be transmitted across a communication channel. Engineers use the quantity $$G(n) =(1 - \sqrt3)a_n -(3 - \sqrt3)a_{n+1} +(3 + \sqrt3)a_{n+2}-(1+ \sqrt3)a_{n+3}$$ to detect noise in the signal. a. Show that if the numbers $a_1, a_2, a_3,...$ are in arithmetic progression, then $G(n) = 0$ for all $n = 1, 2, 3, ...$. b. Show that if $G(n) = 0$ for all $n = 1, 2, 3, ...$, then $a_1, a_2, a_3,...$ is an arithmetic progression. [b]p5.[/b] The Olive View Airline in the remote country of Kuklafrania has decided to use the following rule to establish its air routes: If $A$ and $B$ are two distinct cities, then there is to be an air route connecting $A$ with $B$ either if there is no city closer to $A$ than $B$ or if there is no city closer to $B$ than $A$. No further routes will be permitted. Distances between Kuklafranian cities are never equal. Prove that no city will be connected by air routes to more than ¯ve other cities. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot 4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.