Let $ \{a_n\}_{n=1}^\infty $ be an arithmetic progression with $ a_1 > 0 $ and $ 5\cdot a_{13} = 6\cdot a_{19} $ . What is the smallest integer $ n$ such that $ a_n<0 $?

Let $a_1<a_2<...<a_t$ be $t$ given positive integers where no three form an arithmetic progression. For $k=t,t+1,...$ define $a_{k+1}$ to be the smallest positive integer larger than $a_k$ satisfying the condition that no three of $a_1,a_2,...,a_{k+1}$ form an arithmetic progression. For any $x\in\mathbb{R}^+$ define $A(x)$ to be the number of terms in $\{a_i\}_{i\ge 1}$ that are at most $x$. Show that there exist $c>1$ and $K>0$ such that $A(x)\ge c\sqrt{x}$ for any $x>K$.

Let $ \left( a_n \right) ,\left( b_n \right) $ be two sequences of real numbers from the interval $ (-1,1) $ having the property that $$ \max\left( \left| a_{n+1} -a_n \right| ,\left| b_{n+1} -b_n \right| \right) \le\frac{1}{(n+4)(n+5)} , $$ for any natural number. Prove that $ \left| a_nb_n -a_1b_1 \right|\le 1/2, $ for any natural number $ n. $ [i]Cristinel Mortici[/i]

How many primes less than $100$ have $7$ as the ones digit? (Assume the usual base ten representation) $\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ 8$

Let $f(n)$ be the sum of the first $n$ terms of the sequence $0,1,1,2,2,3,3,4, \dots,$ where the $n$th term is given by $$a_n= \begin{cases} n/2 & \text{if } n \text{ is even,} \\ (n-1)/2 & \text{if } n \text{ is odd.} \end{cases}$$ Show that if $x$ and $y$ are positive integers and $x>y$ then $xy=f(x+y)-f(x-y)$.

An integer sequence $a_1,a_2,\ldots,a_n$ has $a_1=0$, $a_n\leq 10$ and $a_{i+1}-a_i\geq 2$ for $1\leq i<n$. How many possibilities are there for this sequence? The sequence may be of any length.

Let $a_1,a_2,...$ and $b_1,b_2,..$. be two arbitrary infinite sequences of natural numbers. Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$.

Four siblings ordered an extra large pizza. Alex ate $\frac15$, Beth $\frac13$, and Cyril $\frac14$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed? $\textbf{(A) } \text{Alex, Beth, Cyril, Dan}$ $\textbf{(B) } \text{Beth, Cyril, Alex, Dan}$ $\textbf{(C) } \text{Beth, Cyril, Dan, Alex}$ $\textbf{(D) } \text{Beth, Dan, Cyril, Alex}$ $\textbf{(E) } \text{Dan, Beth, Cyril, Alex}$

Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

Let $a_0$, $a_1$, $a_2$, $\dots$ be an infinite sequence of real numbers such that $a_0 = \frac{4}{5}$ and \[ a_{n} = 2 a_{n-1}^2 - 1 \] for every positive integer $n$. Let $c$ be the smallest number such that for every positive integer $n$, the product of the first $n$ terms satisfies the inequality \[ a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}. \] What is the value of $100c$, rounded to the nearest integer?

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

[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 high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that the sequence $S_n=a_1+a_2+\ldots+a_n$ is upperbounded and lowerbounded and find its limit as $n\to\infty$.