id stringlengths 10 13 | question stringlengths 18 1.21k | answer stringlengths 1 18 |
|---|---|---|
omnimath-200 | Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and - signs before each of the integers, yielding an expression $+1-2+3-4+5-6+7-8+9-11+12-\cdots-199$. What is the value of the resulting expression? | -100 |
omnimath-201 | In how many different places in the $xy$-plane can a third point, $R$, be placed so that $PQ = QR = PR$ if points $P$ and $Q$ are two distinct points in the $xy$-plane? | 2 |
omnimath-202 | A sequence of real numbers $a_{0}, a_{1}, \ldots$ is said to be good if the following three conditions hold. (i) The value of $a_{0}$ is a positive integer. (ii) For each non-negative integer $i$ we have $a_{i+1}=2 a_{i}+1$ or $a_{i+1}=\frac{a_{i}}{a_{i}+2}$. (iii) There exists a positive integer $k$ such that $a_{k}=2... | 60 |
omnimath-203 | There are 2017 jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of th... | 2014 |
omnimath-204 | Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\frac{2}{3}$. Suppose that $x_{n}=\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \geq 4$. Find the least $n$ such that $x_{n} \leq \frac{1}{10^{6}}$. | 13 |
omnimath-205 | What is the value of \((-1)^{3}+(-1)^{2}+(-1)\)? | -1 |
omnimath-206 | Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 201... | 18 |
omnimath-207 | Let $x, y, n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^{n}-y^{n}=2^{100}$ ? | 49 |
omnimath-208 | How many of the 20 perfect squares $1^{2}, 2^{2}, 3^{2}, \ldots, 19^{2}, 20^{2}$ are divisible by 9? | 6 |
omnimath-209 | For each positive integer $1 \leq m \leq 10$, Krit chooses an integer $0 \leq a_{m}<m$ uniformly at random. Let $p$ be the probability that there exists an integer $n$ for which $n \equiv a_{m}(\bmod m)$ for all $m$. If $p$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100... | 1540 |
omnimath-210 | When the three-digit positive integer $N$ is divided by 10, 11, or 12, the remainder is 7. What is the sum of the digits of $N$? | 19 |
omnimath-211 | Six soccer teams are competing in a tournament in Waterloo. Every team is to play three games, each against a different team. How many different schedules are possible? | 70 |
omnimath-212 | If points $P, Q, R$, and $S$ are arranged in order on a line segment with $P Q=1, Q R=2 P Q$, and $R S=3 Q R$, what is the length of $P S$? | 9 |
omnimath-213 | Let $g_{1}(x)=\frac{1}{3}\left(1+x+x^{2}+\cdots\right)$ for all values of $x$ for which the right hand side converges. Let $g_{n}(x)=g_{1}\left(g_{n-1}(x)\right)$ for all integers $n \geq 2$. What is the largest integer $r$ such that $g_{r}(x)$ is defined for some real number $x$ ? | 5 |
omnimath-214 | Fifty numbers have an average of 76. Forty of these numbers have an average of 80. What is the average of the other ten numbers? | 60 |
omnimath-215 | Three not necessarily distinct positive integers between 1 and 99, inclusive, are written in a row on a blackboard. Then, the numbers, without including any leading zeros, are concatenated to form a new integer $N$. For example, if the integers written, in order, are 25, 6, and 12, then $N=25612$ (and not $N=250612$). ... | 825957 |
omnimath-216 | The average (mean) of a list of 10 numbers is 17. When one number is removed from the list, the new average is 16. What number was removed? | 26 |
omnimath-217 | How many roots does $\arctan x=x^{2}-1.6$ have, where the arctan function is defined in the range $-\frac{p i}{2}<\arctan x<\frac{p i}{2}$ ? | 2 |
omnimath-218 | What is the integer formed by the rightmost two digits of the integer equal to \(4^{127} + 5^{129} + 7^{131}\)? | 52 |
omnimath-219 | Last Thursday, each of the students in M. Fermat's class brought one piece of fruit to school. Each brought an apple, a banana, or an orange. In total, $20\%$ of the students brought an apple and $35\%$ brought a banana. If 9 students brought oranges, how many students were in the class? | 20 |
omnimath-220 | Let $a, b, c, d, e$ be nonnegative integers such that $625 a+250 b+100 c+40 d+16 e=15^{3}$. What is the maximum possible value of $a+b+c+d+e$ ? | 153 |
omnimath-221 | We are given some similar triangles. Their areas are $1^{2}, 3^{2}, 5^{2} \ldots$, and $49^{2}$. If the smallest triangle has a perimeter of 4, what is the sum of all the triangles' perimeters? | 2500 |
omnimath-222 | Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \] | 120 |
omnimath-223 | The integers $1,2,4,5,6,9,10,11,13$ are to be placed in the circles and squares below with one number in each shape. Each integer must be used exactly once and the integer in each circle must be equal to the sum of the integers in the two neighbouring squares. If the integer $x$ is placed in the leftmost square and the... | 20 |
omnimath-224 | Compute the sum of all positive real numbers \(x \leq 5\) satisfying \(x=\frac{\left\lceil x^{2}\right\rceil+\lceil x\rceil \cdot\lfloor x\rfloor}{\lceil x\rceil+\lfloor x\rfloor}\). | 85 |
omnimath-225 | $S$ is a set of complex numbers such that if $u, v \in S$, then $u v \in S$ and $u^{2}+v^{2} \in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most 1 is finite. What is the largest possible value of $N$ ? | 13 |
omnimath-226 | In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Com... | 1 |
omnimath-227 | Determine which of the following expressions has the largest value: $4^2$, $4 \times 2$, $4 - 2$, $\frac{4}{2}$, or $4 + 2$. | 16 |
omnimath-228 | Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a card with greater value gets a poi... | 2982 |
omnimath-229 | Given positive integers \(a_{1}, a_{2}, \ldots, a_{2023}\) such that \(a_{k}=\sum_{i=1}^{2023}\left|a_{k}-a_{i}\right|\) for all \(1 \leq k \leq 2023\), find the minimum possible value of \(a_{1}+a_{2}+\cdots+a_{2023}\). | 2046264 |
omnimath-230 | Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed? | 8 |
omnimath-231 | Compute the number of sequences of integers $(a_{1}, \ldots, a_{200})$ such that the following conditions hold. - $0 \leq a_{1}<a_{2}<\cdots<a_{200} \leq 202$. - There exists a positive integer $N$ with the following property: for every index $i \in\{1, \ldots, 200\}$ there exists an index $j \in\{1, \ldots, 200\}$ suc... | 20503 |
omnimath-232 | Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $. | 24 |
omnimath-233 | Find the maximum number of points $X_{i}$ such that for each $i$, $\triangle A B X_{i} \cong \triangle C D X_{i}$. | 4 |
omnimath-234 | What is the minimum value of the product $\prod_{i=1}^{6} \frac{a_{i}-a_{i+1}}{a_{i+2}-a_{i+3}}$ given that $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}\right)$ is a permutation of $(1,2,3,4,5,6)$? | 1 |
omnimath-235 | The pairwise products $a b, b c, c d$, and $d a$ of positive integers $a, b, c$, and $d$ are $64,88,120$, and 165 in some order. Find $a+b+c+d$. | 42 |
omnimath-236 | What is the value of the expression \( 4 + \frac{3}{10} + \frac{9}{1000} \)? | 4.309 |
omnimath-237 | Define the sequence $\{x_{i}\}_{i \geq 0}$ by $x_{0}=2009$ and $x_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k}$ for all $n \geq 1$. Compute the value of $\sum_{n=0}^{2009} 2^{n} x_{n}$ | 2009 |
omnimath-238 | In a 3 by 3 grid of unit squares, an up-right path is a path from the bottom left corner to the top right corner that travels only up and right in steps of 1 unit. For such a path $p$, let $A_{p}$ denote the number of unit squares under the path $p$. Compute the sum of $A_{p}$ over all up-right paths $p$. | 90 |
omnimath-239 | Eight celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If order does not matter, how many different lists are possible? | 3507 |
omnimath-240 | A lock code is made up of four digits that satisfy the following rules: - At least one digit is a 4, but neither the second digit nor the fourth digit is a 4. - Exactly one digit is a 2, but the first digit is not 2. - Exactly one digit is a 7. - The code includes a 1, or the code includes a 6, or the code includes two... | 22 |
omnimath-241 | Find out the maximum value of the numbers of edges of a solid regular octahedron that we can see from a point out of the regular octahedron.(We define we can see an edge $AB$ of the regular octahedron from point $P$ outside if and only if the intersection of non degenerate triangle $PAB$ and the solid regular octahedro... | 9 |
omnimath-242 | In a magic square, what is the sum \( a+b+c \)? | 47 |
omnimath-243 | A convex 2019-gon \(A_{1}A_{2}\ldots A_{2019}\) is cut into smaller pieces along its 2019 diagonals of the form \(A_{i}A_{i+3}\) for \(1 \leq i \leq 2019\), where \(A_{2020}=A_{1}, A_{2021}=A_{2}\), and \(A_{2022}=A_{3}\). What is the least possible number of resulting pieces? | 5049 |
omnimath-244 | A function $f:\{1,2,3,4,5\} \rightarrow\{1,2,3,4,5\}$ is said to be nasty if there do not exist distinct $a, b \in\{1,2,3,4,5\}$ satisfying $f(a)=b$ and $f(b)=a$. How many nasty functions are there? | 1950 |
omnimath-245 | The English alphabet, which has 26 letters, is randomly permuted. Let \(p_{1}\) be the probability that \(\mathrm{AB}, \mathrm{CD}\), and \(\mathrm{EF}\) all appear as contiguous substrings. Let \(p_{2}\) be the probability that \(\mathrm{ABC}\) and \(\mathrm{DEF}\) both appear as contiguous substrings. Compute \(\frac... | 23 |
omnimath-246 | You are given a $10 \times 2$ grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent? | 36 |
omnimath-247 | The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice ... | 13 |
omnimath-248 | Let $f:X\rightarrow X$, where $X=\{1,2,\ldots ,100\}$, be a function satisfying:
1) $f(x)\neq x$ for all $x=1,2,\ldots,100$;
2) for any subset $A$ of $X$ such that $|A|=40$, we have $A\cap f(A)\neq\emptyset$.
Find the minimum $k$ such that for any such function $f$, there exist a subset $B$ of $X$, where $|B|=k$, such ... | 69 |
omnimath-249 | In a group of people, there are 13 who like apples, 9 who like blueberries, 15 who like cantaloupe, and 6 who like dates. (A person can like more than 1 kind of fruit.) Each person who likes blueberries also likes exactly one of apples and cantaloupe. Each person who likes cantaloupe also likes exactly one of blueberri... | 22 |
omnimath-250 | In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at $(0,0)$ and the line segment from $(-1,0)$ to $(1,0)$. David stands at the point $(-1,0)$ and shines a flashlight into the room at an angle of $46^{\circ}$ above the horizontal. How many times do... | 65 |
omnimath-251 | Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$. | 120 |
omnimath-252 | How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row? | 199776 |
omnimath-253 | For a positive integer $n$, let $p(n)$ denote the product of the positive integer factors of $n$. Determine the number of factors $n$ of 2310 for which $p(n)$ is a perfect square. | 27 |
omnimath-254 | How many ways are there to insert +'s between the digits of 111111111111111 (fifteen 1's) so that the result will be a multiple of 30? | 2002 |
omnimath-255 | Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square. | 179 |
omnimath-256 | What is the perimeter of $\triangle UVZ$ if $UVWX$ is a rectangle that lies flat on a horizontal floor, a vertical semi-circular wall with diameter $XW$ is constructed, point $Z$ is the highest point on this wall, and $UV=20$ and $VW=30$? | 86 |
omnimath-257 | For how many positive integers $n$, with $n \leq 100$, is $n^{3}+5n^{2}$ the square of an integer? | 8 |
omnimath-258 | Compute the number of real solutions $(x, y, z, w)$ to the system of equations: $$\begin{array}{rlrl} x & =z+w+z w x & z & =x+y+x y z \\ y & =w+x+w x y & w & =y+z+y z w \end{array}$$ | 5 |
omnimath-259 | Find the number of real zeros of $x^{3}-x^{2}-x+2$. | 1 |
omnimath-260 | Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is $3, 14, 25, 36, \ldots$. What is a number that will appear in Ewan's sequence? | 113 |
omnimath-261 | Find the largest integer $n$ such that the following holds: there exists a set of $n$ points in the plane such that, for any choice of three of them, some two are unit distance apart. | 7 |
omnimath-262 | How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon? | 12 |
omnimath-263 | Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\left|x_{2012}\right|$ ? | 84 |
omnimath-264 | Stan has a stack of 100 blocks and starts with a score of 0, and plays a game in which he iterates the following two-step procedure: (a) Stan picks a stack of blocks and splits it into 2 smaller stacks each with a positive number of blocks, say $a$ and $b$. (The order in which the new piles are placed does not matter.)... | 4950 |
omnimath-265 | Let $F(0)=0, F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2} F(n-1)-F(n-2)$ for $n \geq 2$. Determine whether or not $\sum_{n=0}^{\infty} \frac{1}{F\left(2^{n}\right)}$ is a rational number. | 1 |
omnimath-266 | Suppose there exists a convex $n$-gon such that each of its angle measures, in degrees, is an odd prime number. Compute the difference between the largest and smallest possible values of $n$. | 356 |
omnimath-267 | For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ good if $\varphi(n)+4 \tau(n)=n$. For example, the number 44 is good because... | 172 |
omnimath-268 | Narsa buys a package of 45 cookies on Monday morning. How many cookies are left in the package after Friday? | 15 |
omnimath-269 | If $(2)(3)(4) = 6x$, what is the value of $x$? | 4 |
omnimath-270 | What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with \[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\] | 4 |
omnimath-271 | If $x=1$ is a solution of the equation $x^{2} + ax + 1 = 0$, what is the value of $a$? | -2 |
omnimath-272 | Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are diff... | 222480 |
omnimath-273 | Kelvin the Frog and 10 of his relatives are at a party. Every pair of frogs is either friendly or unfriendly. When 3 pairwise friendly frogs meet up, they will gossip about one another and end up in a fight (but stay friendly anyway). When 3 pairwise unfriendly frogs meet up, they will also end up in a fight. In all ot... | 28 |
omnimath-274 | Square $P Q R S$ has an area of 900. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $P S$. What is the area of triangle $P M N$? | 112.5 |
omnimath-275 | Two integers are relatively prime if they don't share any common factors, i.e. if their greatest common divisor is 1. Define $\varphi(n)$ as the number of positive integers that are less than $n$ and relatively prime to $n$. Define $\varphi_{d}(n)$ as the number of positive integers that are less than $d n$ and relativ... | 41 |
omnimath-276 | An 8 by 8 grid of numbers obeys the following pattern: 1) The first row and first column consist of all 1s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i-1)$ by $(j-1)$ sub-grid with row less than $i$ and column less than $j$. What is the number in the 8th row and 8th column? | 2508 |
omnimath-277 | How many ways are there to color every integer either red or blue such that \(n\) and \(n+7\) are the same color for all integers \(n\), and there does not exist an integer \(k\) such that \(k, k+1\), and \(2k\) are all the same color? | 6 |
omnimath-278 | What is the smallest integer that can be placed in the box so that $\frac{1}{2} < \frac{\square}{9}$? | 5 |
omnimath-279 | Determine the positive real value of $x$ for which $$\sqrt{2+A C+2 C x}+\sqrt{A C-2+2 A x}=\sqrt{2(A+C) x+2 A C}$$ | 4 |
omnimath-280 | Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M$, or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order. | 1076 |
omnimath-281 | Let $a, b, c$ be the three roots of $p(x)=x^{3}+x^{2}-333 x-1001$. Find $a^{3}+b^{3}+c^{3}$. | 2003 |
omnimath-282 | If $u=-6$ and $x= rac{1}{3}(3-4 u)$, what is the value of $x$? | 9 |
omnimath-283 | A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. What is the sum when every second term is added up, starting with the first term and ending with the second last term? | 2151 |
omnimath-284 | If $a, b, x$, and $y$ are real numbers such that $a x+b y=3, a x^{2}+b y^{2}=7, a x^{3}+b y^{3}=16$, and $a x^{4}+b y^{4}=42$, find $a x^{5}+b y^{5}$ | 20 |
omnimath-285 | Randall proposes a new temperature system called Felsius temperature with the following conversion between Felsius \(^{\circ} \mathrm{E}\), Celsius \(^{\circ} \mathrm{C}\), and Fahrenheit \(^{\circ} \mathrm{F}\): \(^{\circ} E=\frac{7 \times{ }^{\circ} \mathrm{C}}{5}+16=\frac{7 \times{ }^{\circ} \mathrm{F}-80}{9}\). For... | -120 |
omnimath-286 | A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$. | 1223456 |
omnimath-287 | If $x=3$, $y=2x$, and $z=3y$, what is the value of $z$? | 18 |
omnimath-288 | Let $S$ be the sum of all the real coefficients of the expansion of $(1+i x)^{2009}$. What is $\log _{2}(S)$ ? | 1004 |
omnimath-289 | Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain? | 94 |
omnimath-290 | The Athenas are playing a 44 game season. They have 20 wins and 15 losses so far. What is the smallest number of their remaining games that they must win to make the playoffs, given they must win at least 60% of all of their games? | 7 |
omnimath-291 | An ant starts at the point $(0,0)$ in the Cartesian plane. In the first minute, the ant faces towards $(1,0)$ and walks one unit. Each subsequent minute, the ant chooses an angle $\theta$ uniformly at random in the interval $\left[-90^{\circ}, 90^{\circ}\right]$, and then turns an angle of $\theta$ clockwise (negative ... | 45 |
omnimath-292 | Define the sequence of positive integers $\left\{a_{n}\right\}$ as follows. Let $a_{1}=1, a_{2}=3$, and for each $n>2$, let $a_{n}$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding 2 (in base $n$). For example, $a_{2}=3_{10}=11_{2}$, so $a_{3}=11_{3}+2_{3}... | 23097 |
omnimath-293 | If $\left(a+\frac{1}{a}\right)^{2}=3$, find $\left(a+\frac{1}{a}\right)^{3}$ in terms of $a$. | 0 |
omnimath-294 | In a single-elimination tournament consisting of $2^{9}=512$ teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy is then given the resu... | 45 |
omnimath-295 | Kimothy starts in the bottom-left square of a 4 by 4 chessboard. In one step, he can move up, down, left, or right to an adjacent square. Kimothy takes 16 steps and ends up where he started, visiting each square exactly once (except for his starting/ending square). How many paths could he have taken? | 12 |
omnimath-296 | We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)$ for all positive integers $n$. Find the sum of all good numbers. | 296 |
omnimath-297 | Let $S$ be a subset of the set $\{1,2,3, \ldots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a-b$ does not divide the sum $a+b$. Find the maximum possible size of $S$. | 672 |
omnimath-298 | On a blackboard a stranger writes the values of $s_{7}(n)^{2}$ for $n=0,1, \ldots, 7^{20}-1$, where $s_{7}(n)$ denotes the sum of digits of $n$ in base 7 . Compute the average value of all the numbers on the board. | 3680 |
omnimath-299 | The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$? | 47 |
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