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We call a positive integer double pandigital if it uses all the digits 0 to 9 exactly twice (with no leading zero). For example, 40561817703823564929 is one such number. How many double pandigital numbers are divisible by 11? | 25,500 |
Define the sequence a_1, a_2, a_3, \dots as: a_1 = 1 a_{n+1} = 6a_n^2 + 10a_n + 3 for n \ge 1. Examples: a_3 = 2359 a_6 = 269221280981320216750489044576319 a_6 \bmod 1\,000\,000\,007 = 203064689 a_{100} \bmod 1\,000\,000\,007 = 456482974 Define B(x,y,n) as \sum (a_n \bmod p) for every prime p such that x \le p \le x+y.... | 25,501 |
70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours. What is the expected number of distinct colours in 20 randomly picked balls? Give your answer with nine digits after the decimal point (a.bcdefghij). | 25,502 |
The Collatz sequence is defined as:
a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.. The Collatz conjecture states that starting from any positive integer, the sequence eventually reaches the cycle 1,4,2,1, \dots. We shall define the sequen... | 25,503 |
Let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers. For example, W(144,4) = 7. There are 7 ways in which 144 can be written as a product of 4 distinct positive integers: 144 = 1 \times 2 \times 4 \times 18 144 = 1 \times 2 \times 8 \times 9 144 = 1 \times 2 \times ... | 25,504 |
Given an integer sided triangle ABC: Let I be the incenter of ABC. Let D be the intersection between the line AI and the circumcircle of ABC (A \ne D). We define F(L) as the sum of BC for the triangles ABC that satisfy AC = DI and BC \le L. For example, F(15) = 45 because the triangles ABC with (BC,AC,AB) = (6,4,5), (1... | 25,505 |
Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods and disks of different sizes that can slide onto any of the rods. The game begins with a stack of n disks placed on the leftmost rod in descending order by size. The objective of the game is to move all... | 25,506 |
For positive integers n and m, we define two polynomials F_n(x) = x^n and G_m(x) = (x-1)^m. We also define a polynomial R_{n,m}(x) as the remainder of the division of F_n(x) by G_m(x). For example, R_{6,3}(x) = 15x^2 - 24x + 10. Let C(n, m, d) be the absolute value of the coefficient of the d-th degree term of R_{n,m}(... | 25,507 |
A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games. Each game costs m pounds to play and starts with an initial pot of 1 pound. The gambler flips an unbiased coin. Every time a head appears, the pot is doubled and the gambler continues. When a tail app... | 25,508 |
The number of divisors of 120 is 16. In fact 120 is the smallest number having 16 divisors. Find the smallest number with 2^{500500} divisors. Give your answer modulo 500500507. | 25,509 |
The eight divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
The ten numbers not exceeding 100 having exactly eight divisors are 24, 30, 40, 42, 54, 56, 66, 70, 78 and 88.
Let f(n) be the count of numbers not exceeding n with exactly eight divisors. You are given f(100) = 10, f(1000) = 180 and f(10^6) = 224427. Find f(10^... | 25,510 |
We define a block to be a rectangle with a height of 1 and an integer-valued length. Let a castle be a configuration of stacked blocks. Given a game grid that is w units wide and h units tall, a castle is generated according to the following rules: Blocks can be placed on top of other blocks as long as nothing sticks o... | 25,511 |
Alice is playing a game with n cards numbered 1 to n. A game consists of iterations of the following steps. (1) Alice picks one of the cards at random. (2) Alice cannot see the number on it. Instead, Bob, one of her friends, sees the number and tells Alice how many previously-seen numbers are bigger than the number whi... | 25,512 |
Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows: A(a, 0), B(0, b), C(-c, 0), D(0, -d), where 1 \le a, b, c, d \le m and a, b, c, d, m are integers. It can be shown that for m = 4 there are exactly 256 valid ways to construct ABCD. Of these 256 quadrilaterals, 42 of ... | 25,513 |
Let: \begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text { } \rfloor \text { is the floor function} \\ x(2k+1)&=(2x(k)+3x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \\ y_n(k)&=\left\{{\begi... | 25,514 |
Consider the infinite repeating sequence of digits: 1234321234321234321... Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the digits in the n-th value is n. The sequence goes as follows: 1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, ... Let v_n be the n-th value in ... | 25,515 |
Let t_n be the tribonacci numbers defined as: t_0 = t_1 = 0; t_2 = 1; t_n = t_{n-1} + t_{n-2} + t_{n-3} for n \ge 3 and let r_n = t_n \text{ mod } 10^7. For each pair of Vectors V_n=(v_1,v_2,v_3) and W_n=(w_1,w_2,w_3) with v_1=r_{12n-11}-r_{12n-10}, v_2=r_{12n-9}+r_{12n-8}, v_3=r_{12n-7} \cdot r_{12n-6} and w_1=r_{12n... | 25,516 |
Consider the Gaussian integer i-1. A base i-1 representation of a Gaussian integer a+bi is a finite sequence of digits d_{n - 1}d_{n - 2}\cdots d_1 d_0 such that: a+bi = d_{n - 1}(i - 1)^{n - 1} + d_{n - 2}(i - 1)^{n - 2} + \cdots + d_1(i - 1) + d_0 Each d_k is in \{0,1\} There are no leading zeroes, i.e. d_{n-1} \ne 0... | 25,517 |
Anton and Bertrand love to play three pile Nim. However, after a lot of games of Nim they got bored and changed the rules somewhat. They may only take a number of stones from a pile that is a proper divisor a proper divisor of n is a divisor of n smaller than n of the number of stones present in the pile. E.g. if a pil... | 25,518 |
Circles A and B are tangent to each other and to line L at three distinct points. Circle C is inside the space between A, B and L, and tangent to all three. Let r_A, r_B and r_C be the radii of A, B and C respectively. Let S(n) = \sum r_A + r_B + r_C, for 0 \lt r_A \le r_B \le n where r_A, r_B and r_C are integers.
The... | 25,519 |
Let Seq(n,k) be the number of positive-integer sequences \{a_i\}_{1 \le i \le n} of length n such that: n is divisible by a_i for 1 \le i \le n, and n + a_1 + a_2 + \cdots + a_n is divisible by k. Examples: Seq(3,4) = 4, and the 4 sequences are: \{1, 1, 3\} \{1, 3, 1\} \{3, 1, 1\} \{3, 3, 3\} Seq(4,11) = 8, and the 8 s... | 25,520 |
Let \varphi(n) be Euler's totient function. Let f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \bmod (n+1). Let g(n)=\sum_{i=1}^{n} f(i). g(100)=2007. Find g(5 \times 10^8). | 25,521 |
ABC is an integral sided triangle with sides a \le b \le c. m_C is the median connecting C and the midpoint of AB. F(n) is the number of such triangles with c \le n for which m_C has integral length as well. F(10)=3 and F(50)=165. Find F(100000). | 25,522 |
A geoboard (of order N) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates 0 \le x, y \le N. John begins with a pinless geoboard. Each position on the board is a hole that can be filled with a pin. John decides to generate a random integer betwe... | 25,523 |
Let d(p, n, 0) be the multiplicative inverse of n modulo prime p, defined as n \times d(p, n, 0) = 1 \bmod p. Let d(p, n, k) = \sum_{i = 1}^n d(p, i, k - 1) for k \ge 1. Let D(a, b, k) = \sum (d(p, p-1, k) \bmod p) for all primes a \le p \lt a + b. You are given: D(101,1,10) = 45 D(10^3,10^2,10^2) = 8334 D(10^6,10^3,10... | 25,524 |
5-smooth numbers are numbers whose largest prime factor doesn't exceed 5. 5-smooth numbers are also called Hamming numbers. Let S(L) be the sum of the numbers n not exceeding L such that Euler's totient function \phi(n) is a Hamming number. S(100)=3728. Find S(10^{12}). Give your answer modulo 2^{32}. | 25,525 |
For every real number a \gt 1 is given the sequence g_a by: g_{a}(x)=1 for x \lt a g_{a}(x)=g_{a}(x-1)+g_a(x-a) for x \ge a G(n)=g_{\sqrt {n}}(n) G(90)=7564511. Find \sum G(p) for p prime and 10000000 \lt p \lt 10010000 Give your answer modulo 1000000007. | 25,526 |
Let S(n) = \sum a + b + c over all triples (a, b, c) such that: a, b and c are prime numbers. a \lt b \lt c \lt n. a+1, b+1, and c+1 form a geometric sequence . For example, S(100) = 1035 with the following triples: (2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 11, 17), (7, 23, 71), (11, 23, 47), (17, 23, 31),... | 25,527 |
An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly two coins in the row below is called a fountain of coins. Let f(n) be the number of possible fountains with n coins. For 4 coins there are three possible arrangements: Therefore f(4... | 25,528 |
We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times. For example, 141221242 is a 9-digit simber because it has three 1's, four 2's and two 4's. Let Q(n) be the count of all simbers with at most n di... | 25,529 |
Let \operatorname{smpf}(n) be the smallest prime factor of n. \operatorname{smpf}(91)=7 because 91=7\times 13 and \operatorname{smpf}(45)=3 because 45=3\times 3\times 5. Let S(n) be the sum of \operatorname{smpf}(i) for 2 \le i \le n. E.g. S(100)=1257. Find S(10^{12}) \bmod 10^9. | 25,530 |
Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, instead. To cut costs, Hilbert wished to power the new hotel with his own special generator. Each floor would send power to the floor above it, with the top floor sending power back down to the bottom floo... | 25,531 |
Consider the following algorithm for sorting a list: 1. Starting from the beginning of the list, check each pair of adjacent elements in turn. 2. If the elements are out of order: a. Move the smallest element of the pair at the beginning of the list. b. Restart the process from step 1. 3. If all pairs are in order, sto... | 25,532 |
Consider the following algorithm for sorting a list: 1. Starting from the beginning of the list, check each pair of adjacent elements in turn. 2. If the elements are out of order: a. Move the smallest element of the pair at the beginning of the list. b. Restart the process from step 1. 3. If all pairs are in order, sto... | 25,533 |
An ellipse E(a, b) is given at its initial position by equation: \frac {x^2} {a^2} + \frac {(y - b)^2} {b^2} = 1 The ellipse rolls without slipping along the x axis for one complete turn. Interestingly, the length of the curve generated by a focus is independent from the size of the minor axis: F(a,b) = 2 \pi \max(a,b... | 25,534 |
Let f(n) be the largest prime factor of n. Let g(n) = f(n) + f(n + 1) + f(n + 2) + f(n + 3) + f(n + 4) + f(n + 5) + f(n + 6) + f(n + 7) + f(n + 8), the sum of the largest prime factor of each of nine consecutive numbers starting with n. Let h(n) be the maximum value of g(k) for 2 \le k \le n. You are given: f(100) = 5 ... | 25,535 |
A secret integer t is selected at random within the range 1 \le t \le n. The goal is to guess the value of t by making repeated guesses, via integer g. After a guess is made, there are three possible outcomes, in which it will be revealed that either g \lt t, g = t, or g \gt t. Then the process can repeat as necessary.... | 25,536 |
Let S(n, k, b) represent the number of valid solutions to x_1 + x_2 + \cdots + x_k \le n, where 0 \le x_m \le b^m for all 1 \le m \le k. For example, S(14,3,2) = 135, S(200,5,3) = 12949440, and S(1000,10,5) \bmod 1\,000\,000\,007 = 624839075. Find (\sum_{10 \le k \le 15} S(10^k, k, k)) \bmod 1\,000\,000\,007. | 25,537 |
A 10-substring of a number is a substring of its digits that sum to 10. For example, the 10-substrings of the number 3523014 are: 352 3014 3 523 014 3 5230 14 35 23014 A number is called 10-substring-friendly if every one of its digits belongs to a 10-substring. For example, 3523014 is 10-substring-friendly, but 28546 ... | 25,538 |
Every divisor d of a number n has a complementary divisor n/d. Let f(n) be the sum of the greatest common divisor of d and n/d over all positive divisors d of n, that is
f(n)=\displaystyle\sum_{d\mid n}\gcd(d,\frac n d). Let F be the summatory function of f, that is
F(k)=\displaystyle\sum_{n=1}^k f(n). You are given th... | 25,539 |
Let g(a, n, b, m) be the smallest non-negative solution x to the system: x = a \bmod n x = b \bmod m if such a solution exists, otherwise 0. E.g. g(2,4,4,6)=10, but g(3,4,4,6)=0. Let \phi(n) be Euler's totient function. Let f(n,m)=g(\phi(n),n,\phi(m),m) Find \sum f(n,m) for 1000000 \le n \lt m \lt 1005000. | 25,540 |
Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present. His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a coloured line... | 25,541 |
The Carmichael function \lambda(n) is defined as the smallest positive integer m such that a^m = 1 modulo n for all integers a coprime with n. For example \lambda(8) = 2 and \lambda(240) = 4. Define L(n) as the smallest positive integer m such that \lambda(k) \ge n for all k \ge m. For example, L(6) = 241 and L(100) = ... | 25,542 |
The classical eight queens puzzle is the well known problem of placing eight chess queens on an 8 \times 8 chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of 92 distinct configurations can be found for eight queens. The general case asks for... | 25,543 |
Consider the infinite integer sequence S starting with: S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, \dots Circle the first occurrence of each integer. S = \enclose{circle}1, 1, \enclose{circle}2, 1, \enclose{circle}3, 2, \enclose{circle}4, 1, \enclose{circle}5, 3, \enclose{circle}6, 2, \enclose{ci... | 25,544 |
Let S(n) be the sum of all positive integers m not exceeding n having the following property: a^{m + 4} \equiv a \pmod m for all integers a. The values of m \le 100 that satisfy this property are 1, 2, 3, 5 and 21, thus S(100) = 1+2+3+5+21 = 32. You are given S(10^6) = 22868117. Find S(10^{12}). | 25,545 |
Let \pi(x) be the prime counting function, i.e. the number of prime numbers less than or equal to x. For example,\pi(1)=0, \pi(2)=1, \pi(100)=25. Let T(n, k) be the number of k-tuples (x_1, \dots, x_k) which satisfy: 1. every x_i is a positive integer; 2. \displaystyle \sum_{i=1}^k \pi(x_i)=n For example T(3,3)=19. The... | 25,546 |
Consider a positive integer sequence S = (s_1, s_2, \dots, s_n). Let f(S) be the perimeter of the maximum-area quadrilateral whose side lengths are 4 elements (s_i, s_j, s_k, s_l) of S (all i, j, k, l distinct). If there are many quadrilaterals with the same maximum area, then choose the one with the largest perimeter.... | 25,547 |
Start from an ordered list of all integers from 1 to n. Going from left to right, remove the first number and every other number afterward until the end of the list. Repeat the procedure from right to left, removing the right most number and every other number from the numbers left. Continue removing every other number... | 25,548 |
A Pythagorean triple consists of three positive integers a, b and c satisfying a^2+b^2=c^2. The triple is called primitive if a, b and c are relatively prime. Let P(n) be the number of primitive Pythagorean triples with a \lt b \lt c \le n. For example P(20) = 3, since there are three triples: (3,4,5), (5,12,13) and (8... | 25,549 |
The n th harmonic number H_n is defined as the sum of the multiplicative inverses of the first n positive integers, and can be written as a reduced fraction a_n/b_n. H_n = \displaystyle \sum_{k=1}^n \frac 1 k = \frac {a_n} {b_n}, with \gcd(a_n, b_n)=1. Let M(p) be the largest value of n such that b_n is not divisible b... | 25,550 |
Let S(k) be the sum of three or more distinct positive integers having the following properties: No value exceeds k. The values form a geometric progression . The sum is maximal. S(4) = 4 + 2 + 1 = 7 S(10) = 9 + 6 + 4 = 19 S(12) = 12 + 6 + 3 = 21 S(1000) = 1000 + 900 + 810 + 729 = 3439 Let T(n) = \sum_{k=4}^n (-1)^k S(... | 25,551 |
Define function P(n, k) = 1 if n can be written as the sum of k prime numbers (with repetitions allowed), and P(n, k) = 0 otherwise. For example, P(10,2) = 1 because 10 can be written as either 3 + 7 or 5 + 5, but P(11,2) = 0 because no two primes can sum to 11. Let S(n) be the sum of all P(i,k) over 1 \le i,k \le n. F... | 25,552 |
Let F(r, c, n) be the number of ways to colour a rectangular grid with r rows and c columns using at most n colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent. For example, F(2,2,3) = 18, F(2,2,20) = 130340, and F(3,4,6) = 102923670. Let S(r,... | 25,553 |
The sum of the k th powers of the first n positive integers can be expressed as a polynomial of degree k+1 with rational coefficients, the Faulhaber's Formulas : 1^k + 2^k + ... + n^k = \sum_{i=1}^n i^k = \sum_{i=1}^{k+1} a_{i} n^i = a_{1} n + a_{2} n^2 + ... + a_{k} n^k + a_{k+1} n^{k + 1}, where a_i's are rational co... | 25,554 |
Define f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor) where f_k(0) = 1 and \lfloor x \rfloor denotes the floor function. For example, f_5(10) = 18, f_7(100) = 1003, and f_2(10^3) = 264830889564. Find (\sum_{k=2}^{10} f_k(10^{14})) \bmod (10^9+7). | 25,555 |
Assuming that two points are chosen randomly (with uniform distribution ) within a rectangle, it is possible to determine the expected value of the distance between these two points. For example, the expected distance between two random points in a unit square is about 0.521405, while the expected distance between two ... | 25,556 |
A gozinta chain for n is a sequence \{1,a,b,\dots,n\} where each element properly divides the next. There are eight gozinta chains for 12: \{1,12\}, \{1,2,12\}, \{1,2,4,12\}, \{1,2,6,12\}, \{1,3,12\}, \{1,3,6,12\}, \{1,4,12\} and \{1,6,12\}. Let g(n) be the number of gozinta chains for n, so g(12)=8. g(48)=48 and g(120... | 25,557 |
The smallest number m such that 10 divides m! is m=5. The smallest number m such that 25 divides m! is m=10. Let s(n) be the smallest number m such that n divides m!. So s(10)=5 and s(25)=10. Let S(n) be \sum s(i) for 2 \le i \le n. S(100)=2012. Find S(10^8). | 25,558 |
Two players are playing a game, alternating turns. There are k piles of stones.
On each turn, a player has to choose a pile and replace it with two piles of stones under the following two conditions: Both new piles must have a number of stones more than one and less than the number of stones of the original pile. The n... | 25,559 |
Let a_0, a_1, \dots be an integer sequence defined by: a_0 = 1; for n \ge 1, a_n is the sum of the digits of all preceding terms. The sequence starts with 1, 1, 2, 4, 8, 16, 23, 28, 38, 49, \dots You are given a_{10^6} = 31054319. Find a_{10^{15}}. | 25,560 |
Let A_n be the smallest positive integer satisfying A_n \bmod p_i = i for all 1 \le i \le n, where p_i is the
i-th prime. For example A_2 = 5, since this is the smallest positive solution of the system of equations A_2 \bmod 2 = 1 A_2 \bmod 3 = 2 The system of equations for A_3 adds another constraint. That is, A_3 is ... | 25,561 |
Let P(n) be the set of the first n positive integers \{1, 2, \dots, n\}. Let Q(n) be the set of all the non-empty subsets of P(n). Let R(n) be the set of all the non-empty subsets of Q(n). An element X \in R(n) is a non-empty subset of Q(n), so it is itself a set. From X we can construct a graph as follows: Each elemen... | 25,562 |
On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted king) on an 8 \times 8 board. It can be shown that at most n^2 non-attacking centaurs can be placed on a board of size 2n \times 2n. Let C(n) be the number of ways to place n^2 cen... | 25,563 |
The McCarthy 91 function is defined as follows:
M_{91}(n) =
\begin{cases}
n - 10 & \text{if } n > 100 \\
M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100
\end{cases}
We can generalize this definition by abstracting away the constants into new variables:
M_{m,k,s}(n) =
\begin{cases}
... | 25,564 |
A Gaussian integer is a number z = a + bi where a, b are integers and i^2 = -1. Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which b = 0. A Gaussian integer unit is one for which a^2 + b^2 = 1, i.e. one of 1, i, -1, -i. Let's define a proper Gaussian in... | 25,565 |
A triangle is cut into four pieces by two straight lines, each starting at one vertex and ending on the opposite edge. This results in forming three smaller triangular pieces, and one quadrilateral. If the original triangle has an integral area, it is often possible to choose cuts such that all of the four pieces also... | 25,566 |
Let r be the real root of the equation x^3 = x^2 + 1. Every positive integer can be written as the sum of distinct increasing powers of r. If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique. For example, 3 = r^{-10} + r^{-5}... | 25,567 |
An ascent of a column j in a matrix occurs if the value of column j is smaller than the value of column j + 1 in all rows. Let P(k, r, n) be the number of r \times n matrices with the following properties: The rows are permutations of \{1, 2, 3, \dots, n\}. Numbering the first column as 1, a column ascent occurs at col... | 25,568 |
Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the current size of the pile. Two players remove stones in turn. The player who removes the last stone wins. Let L(n, k) be the number of losing starting positions for the first player,... | 25,569 |
Let S(n) be the number of pairs (a,b) of distinct divisors of n such that a divides b. For n=6 we get the following pairs: (1,2), (1,3), (1,6),( 2,6) and (3,6). So S(6)=5. Let p_m\# be the product of the first m prime numbers, so p_2\# = 2*3 = 6. Let E(m, n) be the highest integer k such that 2^k divides S((p_m\#)^n).... | 25,570 |
Construct triangle ABC such that: Vertices A, B and C are lattice points inside or on the circle of radius r centered at the origin; the triangle contains no other lattice point inside or on its edges; the perimeter is maximum. Let R be the circumradius of triangle ABC and T(r) = R/r. For r = 5, one possible triangle h... | 25,571 |
A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to 25 identical rectangles of metal, whi... | 25,572 |
A line segment of length 2n-3 is randomly split into n segments of integer length (n \ge 3). In the sequence given by this split, the segments are then used as consecutive sides of a convex n-polygon, formed in such a way that its area is maximal. All of the \binom{2n-4} {n-1} possibilities for splitting up the initia... | 25,573 |
Let \sigma(n) be the sum of the divisors of n. E.g. the divisors of 4 are 1, 2 and 4, so \sigma(4)=7. The numbers n not exceeding 20 such that 7 divides \sigma(n) are: 4, 12, 13 and 20, the sum of these numbers being 49. Let S(n, d) be the sum of the numbers i not exceeding n such that d divides \sigma(i). So S(20 , ... | 25,574 |
Adam plays the following game with his birthday cake. He cuts a piece forming a circular sector of 60 degrees and flips the piece upside down, with the icing on the bottom. He then rotates the cake by 60 degrees counterclockwise, cuts an adjacent 60 degree piece and flips it upside down. He keeps repeating this, until ... | 25,575 |
Tom has built a random generator that is connected to a row of n light bulbs. Whenever the random generator is activated each of the n lights is turned on with the probability of \frac 1 2, independently of its former state or the state of the other light bulbs. While discussing with his friend Jerry how to use his gen... | 25,576 |
Tom has built a random generator that is connected to a row of n light bulbs. Whenever the random generator is activated each of the n lights is turned on with the probability of \frac 1 2, independently of its former state or the state of the other light bulbs. While discussing with his friend Jerry how to use his gen... | 25,577 |
A mountain range consists of a line of mountains with slopes of exactly 45^\circ, and heights governed by the prime numbers, p_n. The up-slope of the k th mountain is of height p_{2k - 1}, and the downslope is p_{2k}. The first few foot-hills of this range are illustrated below. Tenzing sets out to climb each one in tu... | 25,578 |
A snowflake of order n is formed by overlaying an equilateral triangle (rotated by 180 degrees) onto each equilateral triangle of the same size in a snowflake of order n-1. A snowflake of order 1 is a single equilateral triangle. Some areas of the snowflake are overlaid repeatedly. In the above picture, blue represents... | 25,579 |
A positive number is pandigital in base b if it contains all digits from 0 to b - 1 at least once when written in base b. An n-super-pandigital number is a number that is simultaneously pandigital in all bases from 2 to n inclusively. For example 978 = 1111010010_2 = 1100020_3 = 33102_4 = 12403_5 is the smallest 5-supe... | 25,580 |
A matrix M is called idempotent if M^2 = M. Let M be a three by three matrix :
M=\begin{pmatrix}
a & b & c\\
d & e & f\\
g &h &i\\
\end{pmatrix}. Let C(n) be the number of idempotent three by three matrices M with integer elements such that -n \le a,b,c,d,e,f,g,h,i \le n. C(1)=164 and C(2)=848. Find C(200). | 25,581 |
n runners in very different training states want to compete in a race. Each one of them is given a different starting number k (1\leq k \leq n) according to the runner's (constant) individual racing speed being v_k=\frac{k}{n}. In order to give the slower runners a chance to win the race, n different starting positions... | 25,582 |
Let q be a prime and A \ge B >0 be two integers with the following properties: A and B have no prime factor in common, that is \gcd(A,B)=1. The product AB is divisible by every prime less than q. It can be shown that, given these conditions, any sum A+B<q^2 and any difference 1<A-B<q^2 has to be a prime number. Thus yo... | 25,583 |
It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided to send a team to investigate, of which you are included. Upon entering the vessel you are met by a friendly holographic figure, Katharina, who explains the purpos... | 25,584 |
A bouncing point moves counterclockwise along a circle with circumference 1 with jumps of constant length l \lt 1, until it hits a gap of length g \lt 1, that is placed in a distance d counterclockwise from the starting point. The gap does not include the starting point, that is g+d \lt 1. Let S(l,g,d) be the sum of th... | 25,585 |
An equilateral triangle with integer side length n \ge 3 is divided into n^2 equilateral triangles with side length 1 as shown in the diagram below. The vertices of these triangles constitute a triangular lattice with \frac{(n+1)(n+2)} 2 lattice points. Let H(n) be the number of all regular hexagons that can be found b... | 25,586 |
Any positive integer can be written as a product of prime powers: p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k}, where p_i are distinct prime integers, a_i \gt 0 and p_i \lt p_j if i \lt j. A decreasing prime power positive integer is one for which a_i \ge a_j if i \lt j. For example, 1, 2, 15=3 \times 5, 3... | 25,587 |
A lattice cube is a cube in which all vertices have integer coordinates. Let C(n) be the number of different lattice cubes in which the coordinates of all vertices range between (and including) 0 and n. Two cubes are hereby considered different if any of their vertices have different coordinates. For example, C(1)=1, C... | 25,588 |
A Hilbert number is any positive integer of the form 4k+1 for integer k\geq 0. We shall define a squarefree Hilbert number as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, 117 is a squarefree Hilbert number, equaling 9\times13. However 6237 is a Hilbert numb... | 25,589 |
A number is p-smooth if it has no prime factors larger than p. Let T be the sequence of triangular numbers, i.e. T(n)=n(n+1)/2. Find the sum of all indices n such that T(n) is 47-smooth. | 25,590 |
Let a, b and c be the sides of an integer sided triangle with one angle of 120 degrees, a \le b \le c and b-a \le 100. Let T(n) be the number of such triangles with c \le n. T(1000)=235 and T(10^8)=1245. Find T(10^{100}). | 25,591 |
A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle. An example of an envelope with integral sides is shown below. Note that to form a sensible envelope, the perpendicular height of the flap (BCD) must be smaller than the height of the rectangle (AB... | 25,592 |
A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone that had landed on their planet. On examining the drone, they uncovered a device that sought the answer for the so called " Birthday Problem ". The description of the problem was as follows: If people ... | 25,593 |
Consider the term \small \sqrt{x+\sqrt{y}+\sqrt{z}} that is representing a nested square root . x, y and z are positive integers and y and z are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of x, y and z the given term can b... | 25,594 |
The number 209 can be expressed as a^2 + 3ab + b^2 in two distinct ways: \qquad 209 = 8^2 + 3\cdot 8\cdot 5 + 5^2 \qquad 209 = 13^2 + 3\cdot13\cdot 1 + 1^2 Let f(n,r) be the number of integers k not exceeding n that can be expressed as k=a^2 + 3ab + b^2, with a \gt b \gt 0 integers, in exactly r different ways. You a... | 25,595 |
A square is drawn around a circle as shown in the diagram below on the left. We shall call the blue shaded region the L-section. A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right. We shall call the orange shaded region a concave triangle . It should be clear that t... | 25,596 |
The coefficients in the expansion of (x+1)^k are called binomial coefficients . Analoguously the coefficients in the expansion of (x^4+x^3+x^2+x+1)^k are called quintinomial coefficients . (quintus= Latin for fifth). Consider the expansion of (x^4+x^3+x^2+x+1)^3: x^{12}+3x^{11}+6x^{10}+10x^9+15x^8+18x^7+19x^6+18x^5+15x... | 25,597 |
Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them to play for longer before one of them wins and they have to go home for tea. The game starts as normal with both dropping a stick simultaneously on the upstream side of a bridge. But rather than the ga... | 25,598 |
Let H(n) denote the number of sets of positive integers such that the least common multiple of the integers in the set equals n. E.g.: The integers in the following ten sets all have a least common multiple of 6: \{2,3\}, \{1,2,3\}, \{6\}, \{1,6\}, \{2,6\}, \{1,2,6\}, \{3,6\}, \{1,3,6\}, \{2,3,6\} and \{1,2,3,6\}. Thus... | 25,599 |
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