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A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize. During an n-day period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (abse... | 25,200 |
Let x be a real number. A best approximation to x for the denominator bound d is a rational number \frac r s in reduced form , with s \le d, such that any rational number which is closer to x than \frac r s has a denominator larger than d: |\frac p q -x | < |\frac r s -x| \Rightarrow q > d For example, the best appro... | 25,201 |
A positive integer n is called squarefree, if no square of a prime divides n, thus 1, 2, 3, 5, 6, 7, 10, 11 are squarefree, but not 4, 8, 9, 12. How many squarefree numbers are there below 2^{50}? | 25,202 |
Consider graphs built with the units A: and B: , where the units are glued along
the vertical edges as in the graph . A configuration of type (a, b, c) is a graph thus built of a units A and b units B, where the graph's vertices are coloured using up to c colours, so that no two adjacent vertices have the same colour. ... | 25,203 |
Let's call an integer sided triangle with exactly one angle of 60 degrees a 60-degree triangle. Let r be the radius of the inscribed circle of such a 60-degree triangle. There are 1234 60-degree triangles for which r \le 100. Let T(n) be the number of 60-degree triangles for which r \le n, so T(100) = 1234, T(1000) = 2... | 25,204 |
Build a triangle from all positive integers in the following way: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 . . . Each positive integer has up to eight neighbours in the tr... | 25,205 |
Given is the function f(x) = \lfloor 2^{30.403243784 - x^2}\rfloor \times 10^{-9} (\lfloor \, \rfloor is the floor-function), the sequence u_n is defined by u_0 = -1 and u_{n + 1} = f(u_n). Find u_n + u_{n + 1} for n = 10^{12}. Give your answer with 9 digits after the decimal point. | 25,206 |
A best approximation to a real number x for the denominator bound d is a rational number \frac r s (in reduced form) with s \le d, so that any rational number \frac p q which is closer to x than \frac r s has q \gt d. Usually the best approximation to a real number is uniquely determined for all denominator bounds. How... | 25,207 |
Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do not overlap. There are four uncovered "gaps" which are to be filled iteratively with more tangent circles. At each iteration, a maximally sized circle is placed in each gap, ... | 25,208 |
We shall define a sqube to be a number of the form, p^2 q^3, where p and q are distinct primes. For example, 200 = 5^2 2^3 or 120072949 = 23^2 61^3. The first five squbes are 72, 108, 200, 392, and 500. Interestingly, 200 is also the first number for which you cannot change any single digit to make a prime; we shall ca... | 25,209 |
For any set A of numbers, let \operatorname{sum}(A) be the sum of the elements of A. Consider the set B = \{1,3,6,8,10,11\}. There are 20 subsets of B containing three elements, and their sums are: \begin{align}
\operatorname{sum}(\{1,3,6\}) &= 10,\\
\operatorname{sum}(\{1,3,8\}) &= 12,\\
\operatorname{sum}(\{1,3,10\})... | 25,210 |
Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards. There is an infinitesimal gap at each vertex of the triangle through which a laser beam may pass. Label the vertices A, B and C. There are 2 ways in which a laser beam may enter vertex C, bounce off 11 s... | 25,211 |
The binomial coefficients \displaystyle \binom n k can be arranged in triangular form, Pascal's triangle, like this: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 ......... It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: 1, 2, 3, 4, 5, 6,... | 25,212 |
A Hamming number is a positive number which has no prime factor larger than 5. So the first few Hamming numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15. There are 1105 Hamming numbers not exceeding 10^8. We will call a positive number a generalised Hamming number of type n, if it has no prime factor larger than n. Hence... | 25,213 |
Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2, 3, 4. Colin has six six-sided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, 6. Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal. What is the probability that Pyramid... | 25,214 |
Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0, where each “_” is a single digit. | 25,215 |
For some positive integers k, there exists an integer partition of the form 4^t = 2^t + k, where 4^t, 2^t, and k are all positive integers and t is a real number. The first two such partitions are 4^1 = 2^1 + 2 and 4^{1.5849625\cdots} = 2^{1.5849625\cdots} + 6. Partitions where t is also an integer are called perfect .... | 25,216 |
A robot moves in a series of one-fifth circular arcs (72^\circ), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot. One of 70932 possible closed paths of 25 arcs starting northward is Given that the robot starts facing North, how many journeys of 70 arcs in length can i... | 25,217 |
A k-input binary truth table is a map from k input bits (binary digits, 0 [false] or 1 [true]) to 1 output bit. For example, the 2-input binary truth tables for the logical \mathbin{\text{AND}} and \mathbin{\text{XOR}} functions are: x y x \mathbin{\text{AND}} y 0 0 0 0 1 0 1 0 0 1 1 1 x y x\mathbin{\text{XOR}}y 0 0 0 ... | 25,218 |
Consider the set S(r) of points (x,y) with integer coordinates satisfying |x| + |y| \le r. Let O be the point (0,0) and C the point (r/4,r/4). Let N(r) be the number of points B in S(r), so that the triangle OBC has an obtuse angle, i.e. the largest angle \alpha satisfies 90^\circ \lt \alpha \lt 180^\circ. So, for exam... | 25,219 |
For a positive integer n, let \sigma_2(n) be the sum of the squares of its divisors. For example,
\sigma_2(10) = 1 + 4 + 25 + 100 = 130. Find the sum of all n, 0 \lt n \lt 64\,000\,000 such that \sigma_2(n) is a perfect square. | 25,220 |
An axis-aligned cuboid , specified by parameters \{(x_0, y_0, z_0), (dx, dy, dz)\}, consists of all points (X,Y,Z) such that x_0 \le X \le x_0 + dx, y_0 \le Y \le y_0 + dy and z_0 \le Z \le z_0 + dz. The volume of the cuboid is the product, dx \times dy \times dz. The combined volume of a collection of cuboids is the... | 25,221 |
A 30 \times 30 grid of squares contains 900 fleas, initially one flea per square. When a bell is rung, each flea jumps to an adjacent square at random (usually 4 possibilities, except for fleas on the edge of the grid or at the corners). What is the expected number of unoccupied squares after 50 rings of the bell? Give... | 25,222 |
Let \phi be Euler's totient function, i.e. for a natural number n,
\phi(n) is the number of k, 1 \le k \le n, for which \gcd(k, n) = 1. By iterating \phi, each positive integer generates a decreasing chain of numbers ending in 1. E.g. if we start with 5 the sequence 5,4,2,1 is generated. Here is a listing of all chains... | 25,223 |
Consider the problem of building a wall out of 2 \times 1 and 3 \times 1 bricks (\text{horizontal} \times \text{vertical} dimensions) such that, for extra strength, the gaps between horizontally-adjacent bricks never line up in consecutive layers, i.e. never form a "running crack". For example, the following 9 \times 3... | 25,224 |
Consider numbers t(n) of the form t(n) = 2n^2 - 1 with n \gt 1. The first such numbers are 7, 17, 31, 49, 71, 97, 127 and 161. It turns out that only 49 = 7 \cdot 7 and 161 = 7 \cdot 23 are not prime. For n \le 10000 there are 2202 numbers t(n) that are prime. How many numbers t(n) are prime for n \le 50\,000\,000? | 25,225 |
A positive integer with k (decimal) digits is called balanced if its first \lceil k/2 \rceil digits sum to the same value as its last \lceil k/2 \rceil digits, where \lceil x \rceil, pronounced ceiling of x, is the smallest integer \ge x, thus \lceil \pi \rceil = 4 and \lceil 5 \rceil = 5. So, for example, all palindro... | 25,226 |
Consider the right angled triangle with sides a=7, b=24 and c=25.
The area of this triangle is 84, which is divisible by the perfect numbers 6 and 28. Moreover it is a primitive right angled triangle as \gcd(a,b)=1 and \gcd(b,c)=1. Also c is a perfect square. We will call a right angled triangle perfect if -it is a pri... | 25,227 |
Let A and B be bit strings (sequences of 0's and 1's). If A is equal to the left most length( A ) bits of B , then A is said to be a prefix of B . For example, 00110 is a prefix of 00110 1001, but not of 00111 or 100110. A prefix-free code of size n is a collection of n distinct bit strings such that no string is a pre... | 25,228 |
Let D 0 be the two-letter string "Fa". For n≥1, derive D n from D n-1 by the string-rewriting rules: "a" → "aRbFR" "b" → "LFaLb" Thus, D 0 = "Fa", D 1 = "FaRbFR", D 2 = "FaRbFRRLFaLbFR", and so on. These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit"... | 25,229 |
We shall call a positive integer A an "Alexandrian integer", if there exist integers p, q, r such that: A = p \cdot q \cdot r
and
\dfrac{1}{A} = \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r}. For example, 630 is an Alexandrian integer (p = 5, q = -7, r = -18).
In fact, 630 is the 6 th Alexandrian integer, the first 6 Ale... | 25,230 |
What is the length of the shortest pipe, of internal radius \pu{50 mm}, that can fully contain 21 balls of radii \pu{30 mm}, \pu{31 mm}, \dots, \pu{50 mm}? Give your answer in micrometres (\pu{10^{-6} m}) rounded to the nearest integer. | 25,231 |
Let us call an integer sided triangle with sides a \le b \le c barely acute if the sides satisfy a^2 + b^2 = c^2 + 1. How many barely acute triangles are there with perimeter \le 25\,000\,000? | 25,232 |
Let us call an integer sided triangle with sides a \le b \le c barely obtuse if the sides satisfy a^2 + b^2 = c^2 - 1. How many barely obtuse triangles are there with perimeter \le 75\,000\,000? | 25,233 |
The sequence 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201, \dots is defined by T_1 = T_2 = T_3 = 1 and T_n = T_{n - 1} + T_{n - 2} + T_{n - 3}. It can be shown that 27 does not divide any terms of this sequence. In fact, 27 is the first odd number with this property. Find the 124 th odd number that does not d... | 25,234 |
The blancmange curve is the set of points (x, y) such that 0 \le x \le 1 and y = \sum \limits_{n = 0}^{\infty} {\dfrac{s(2^n x)}{2^n}}, where s(x) is the distance from x to the nearest integer. The area under the blancmange curve is equal to ½, shown in pink in the diagram below. Let C be the circle with centre \left (... | 25,235 |
The Chase is a game played with two dice and an even number of players. The players sit around a table and the game begins with two opposite players having one die each. On each turn, the two players with a die roll it. If the player rolls 1, then the die passes to the neighbour on the left. If the player rolls 6, then... | 25,236 |
Let S_n be the regular n-sided polygon – or shape – whose vertices
v_k (k = 1, 2, \dots, n) have coordinates: \begin{align}
x_k &= \cos((2k - 1)/n \times 180^\circ)\\
y_k &= \sin((2k - 1)/n \times 180^\circ)
\end{align} Each S_n is to be interpreted as a filled shape consisting of all points on the perimeter and in t... | 25,237 |
Consider the number 3600. It is very special, because \begin{alignat}{2}
3600 &= 48^2 + &&36^2\\
3600 &= 20^2 + 2 \times &&40^2\\
3600 &= 30^2 + 3 \times &&30^2\\
3600 &= 45^2 + 7 \times &&15^2
\end{alignat} Similarly, we find that 88201 = 99^2 + 280^2 = 287^2 + 2 \times 54^2 = 283^2 + 3 \times 52^2 = 197^2 + 7 \times ... | 25,238 |
For any two strings of digits, A and B, we define F_{A, B} to be the sequence (A,B,AB,BAB,ABBAB,\dots) in which each term is the concatenation of the previous two. Further, we define D_{A, B}(n) to be the n th digit in the first term of F_{A, B} that contains at least n digits. Example: Let A=1415926535, B=8979323846. ... | 25,239 |
The binomial coefficient \displaystyle \binom {10} 3 = 120. 120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5, and 2 + 2 + 2 + 3 + 5 = 14. So the sum of the terms in the prime factorisation of \displaystyle \binom {10} 3 is 14. Find the sum of the terms in the prime factorisation of \displaystyle \bin... | 25,240 |
Two players share an unbiased coin and take it in turns to play The Race . On Player 1's turn, the coin is tossed once. If it comes up Heads, then Player 1 scores one point; if it comes up Tails, then no points are scored. On Player 2's turn, a positive integer, T, is chosen by Player 2 and the coin is tossed T times. ... | 25,241 |
Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (N,N). It can be shown that f(10000) = 36. What is the sum of all positive integers N \le 10^{11} such that f(N) = 420? | 25,242 |
For an integer n \ge 4, we define the lower prime square root of n, denoted by \operatorname{lps}(n), as the largest prime \le \sqrt n and the upper prime square root of n, \operatorname{ups}(n), as the smallest prime \ge \sqrt n. So, for example, \operatorname{lps}(4) = 2 = \operatorname{ups}(4), \operatorname{lps}(10... | 25,243 |
Given is the arithmetic-geometric sequence u(k) = (900-3k)r^{k - 1}. Let s(n) = \sum_{k = 1}^n u(k). Find the value of r for which s(5000) = -600\,000\,000\,000. Give your answer rounded to 12 places behind the decimal point. | 25,244 |
Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market: Product 'A' 'B' Beluga Caviar 5248 640 Christmas Cake 1312 1888 Gammon Joint 2624 3776 Vintage Port 5760 3776 Champagne Truffles 3936 5664 Although the suppliers try very hard to ship their goods in perfect condition, there i... | 25,245 |
Let T(n) be the number of tours over a 4 \times n playing board such that: The tour starts in the top left corner. The tour consists of moves that are up, down, left, or right one square. The tour visits each square exactly once. The tour ends in the bottom left corner. The diagram shows one tour over a 4 \times 10 boa... | 25,246 |
Create a sequence of numbers using the "Blum Blum Shub" pseudo-random number generator: \begin{align}
s_0 &= 14025256\\
s_{n + 1} &= s_n^2 \bmod 20300713
\end{align} Concatenate these numbers s_0s_1s_2\cdots to create a string w of infinite length. Then, w = {\color{blue}14025256741014958470038053646\cdots} For a posit... | 25,247 |
A set of disks numbered 1 through 100 are placed in a line in random order. What is the probability that we have a partial derangement such that exactly 22 prime number discs are found away from their natural positions? (Any number of non-prime disks may also be found in or out of their natural positions.) Give your an... | 25,248 |
There are 1111 ways in which five 6-sided dice (sides numbered 1 to 6) can be rolled so that the top three sum to 15. Some examples are: D_1,D_2,D_3,D_4,D_5 = 4,3,6,3,5 D_1,D_2,D_3,D_4,D_5 = 4,3,3,5,6 D_1,D_2,D_3,D_4,D_5 = 3,3,3,6,6 D_1,D_2,D_3,D_4,D_5 = 6,6,3,3,3 In how many ways can twenty 12-sided dice (sides number... | 25,249 |
For a positive integer n, let \sigma(n) be the sum of all divisors of n. For example, \sigma(6) = 1 + 2 + 3 + 6 = 12. A perfect number, as you probably know, is a number with \sigma(n) = 2n. Let us define the perfection quotient of a positive integer as p(n) = \dfrac{\sigma(n)}{n}. Find the sum of all positive integers... | 25,250 |
Given the set \{1,2,\dots,n\}, we define f(n, k) as the number of its k-element subsets with an odd sum of elements. For example, f(5,3) = 4, since the set \{1,2,3,4,5\} has four 3-element subsets having an odd sum of elements, i.e.: \{1,2,4\}, \{1,3,5\}, \{2,3,4\} and \{2,4,5\}. When all three values n, k and f(n, k) ... | 25,251 |
A positive fraction whose numerator is less than its denominator is called a proper fraction. For any denominator, d, there will be d - 1 proper fractions; for example, with d = 12: 1 / 12, 2 / 12, 3 / 12, 4 / 12, 5 / 12, 6 / 12, 7 / 12, 8 / 12, 9 / 12, 10 / 12, 11 / 12. We shall call a fraction that cannot be cancelle... | 25,252 |
You probably know the game Fifteen Puzzle . Here, instead of numbered tiles, we have seven red tiles and eight blue tiles. A move is denoted by the uppercase initial of the direction (Left, Right, Up, Down) in which the tile is slid, e.g. starting from configuration ( S ), by the sequence LULUR we reach the configurati... | 25,253 |
We shall call a fraction that cannot be cancelled down a resilient fraction. Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = \dfrac{4}{11}. The resilience of a number d \gt 1 is then \dfrac{\varphi(d)}{d - 1}, where \var... | 25,254 |
A definition for an ellipse is: Given a circle c with centre M and radius r and a point G such that d(G,M) \lt r, the locus of the points that are equidistant from c and G form an ellipse. The construction of the points of the ellipse is shown below. Given are the points M(-2000,1500) and G(8000,1500). Given is also th... | 25,255 |
Consider the region constrained by 1 \le x and 0 \le y \le 1/x. Let S_1 be the largest square that can fit under the curve. Let S_2 be the largest square that fits in the remaining area, and so on. Let the index of S_n be the pair (\text{left}, \text{below}) indicating the number of squares to the left of S_n and the n... | 25,256 |
The first number n for which \phi(n)=13! is 6227180929. Find the 150\,000 th such number. | 25,257 |
Let S = \{2, 3, 5, \dots, 4999\} be the set of prime numbers less than 5000. Find the number of subsets of S, the sum of whose elements is a prime number. Enter the rightmost 16 digits as your answer. | 25,258 |
Find the number of non-empty subsets of \{1^1, 2^2, 3^3,\dots, 250250^{250250}\}, the sum of whose elements is divisible by 250. Enter the rightmost 16 digits as your answer. | 25,259 |
A triplet of positive integers (a, b, c) is called a Cardano Triplet if it satisfies the condition: \sqrt[3]{a + b \sqrt{c}} + \sqrt[3]{a - b \sqrt{c}} = 1 For example, (2,1,5) is a Cardano Triplet. There exist 149 Cardano Triplets for which a + b + c \le 1000. Find how many Cardano Triplets exist such that a + b + c \... | 25,260 |
Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon). As an example, the image below shows a set of ... | 25,261 |
A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers 1 to 40 in order. Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up t... | 25,262 |
Define f(n) as the sum of the factorials of the digits of n. For example, f(342) = 3! + 4! + 2! = 32. Define sf(n) as the sum of the digits of f(n). So sf(342) = 3 + 2 = 5. Define g(i) to be the smallest positive integer n such that sf(n) = i. Though sf(342) is 5, sf(25) is also 5, and it can be verified that g(5) is 2... | 25,263 |
We define the rounded-square-root of a positive integer n as the square root of n rounded to the nearest integer. The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of n: Let d be the number of digits of the number n. If d is odd, set x_0 = 2 \times 10^{(d-1... | 25,264 |
Tatami are rectangular mats, used to completely cover the floor of a room, without overlap. Assuming that the only type of available tatami has dimensions 1 \times 2, there are obviously some limitations for the shape and size of the rooms that can be covered. For this problem, we consider only rectangular rooms with i... | 25,265 |
Given is an integer sided triangle ABC with sides a \le b \le c.
(AB = c, BC = a and AC = b.) The angular bisectors of the triangle intersect the sides at points E, F and G (see picture below). The segments EF, EG and FG partition the triangle ABC into four smaller triangles: AEG, BFE, CGF and EFG. It can be proven th... | 25,266 |
A sequence is defined as: g_k = 1, for 0 \le k \le 1999 g_k = g_{k-2000} + g_{k - 1999}, for k \ge 2000. Find g_k \bmod 20092010 for k = 10^{18}. | 25,267 |
A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules: Uses the digits 1 through 9, in that order and exactly once each. Any successive digits can be concatenated (for example, using the digits 2, 3 and 4 we obtain the number 234). Only the four usual bin... | 25,268 |
A game is played with three piles of stones and two players. On each player's turn, the player may remove one or more stones from the piles. However, if the player takes stones from more than one pile, then the same number of stones must be removed from each of the selected piles. In other words, the player chooses som... | 25,269 |
Let us call a positive integer k a square-pivot , if there is a pair of integers m \gt 0 and n \ge k, such that the sum of the (m+1) consecutive squares up to k equals the sum of the m consecutive squares from (n+1) on: (k - m)^2 + \cdots + k^2 = (n + 1)^2 + \cdots + (n + m)^2. Some small square-pivots are \mathbf 4: 3... | 25,270 |
The following equation represents the continuous topography of a mountainous region, giving the elevation height above sea level h at any point (x, y):
h = \left(5000 - \frac{x^2 + y^2 + xy}{200} + \frac{25(x + y)}2\right) \cdot e^{-\left|\frac{x^2 + y^2}{1000000} - \frac{3(x + y)}{2000} + \frac 7 {10}\right|}. A mosqu... | 25,271 |
Consider the number 6. The divisors of 6 are: 1,2,3 and 6. Every number from 1 up to and including 6 can be written as a sum of distinct divisors of 6: 1=1, 2=2, 3=1+2, 4=1+3, 5=2+3, 6=6. A number n is called a practical number if every number from 1 up to and including n can be expressed as a sum of distinct divisors ... | 25,272 |
Consider all the triangles having: All their vertices on lattice points Integer coordinates . Circumcentre Centre of the circumscribed circle at the origin O. Orthocentre Point where the three altitudes meet at the point H(5, 0). There are nine such triangles having a perimeter \le 50. Listed and shown in ascending ord... | 25,273 |
2^N binary digits can be placed in a circle so that all the N-digit clockwise subsequences are distinct. For N=3, two such circular arrangements are possible, ignoring rotations: For the first arrangement, the 3-digit subsequences, in clockwise order, are: 000, 001, 010, 101, 011, 111, 110 and 100. Each circular arrang... | 25,274 |
The divisors of 12 are: 1,2,3,4,6 and 12. The largest divisor of 12 that does not exceed the square root of 12 is 3. We shall call the largest divisor of an integer n that does not exceed the square root of n the pseudo square root (\operatorname{PSR}) of n. It can be seen that \operatorname{PSR}(3102)=47. Let p be the... | 25,275 |
You are given a unique investment opportunity. Starting with £1 of capital, you can choose a fixed proportion, f , of your capital to bet on a fair coin toss repeatedly for 1000 tosses. Your return is double your bet for heads and you lose your bet for tails. For example, if f = 1/4, for the first toss you bet £0.25, ... | 25,276 |
It can be verified that there are 23 positive integers less than 1000 that are divisible by at least four distinct primes less than 100. Find how many positive integers less than 10^{16} are divisible by at least four distinct primes less than 100. | 25,277 |
A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0. Define P_n as the polynomial whose coefficients are the digits of n. For example, P_{5703}(x) = 5x^3 + 7x^2 + 3. We can see that: P_n(0) is the last digit of n, P_n(1) is the sum of the digits of n, P_n(10) is n itself. Define Z(k) as the numbe... | 25,278 |
A square piece of paper with integer dimensions N \times N is placed with a corner at the origin and two of its sides along the x- and y-axes. Then, we cut it up respecting the following rules: We only make straight cuts between two points lying on different sides of the square, and having integer coordinates. Two cuts... | 25,279 |
For a positive number n, define S(n) as the sum of the integers x, for which 1 \lt x \lt n and x^3 \equiv 1 \bmod n. When n=91, there are 8 possible values for x, namely: 9, 16, 22, 29, 53, 74, 79, 81. Thus, S(91)=9+16+22+29+53+74+79+81=363. Find S(13082761331670030). | 25,280 |
For a positive number n, define C(n) as the number of the integers x, for which 1 \lt x \lt n and x^3 \equiv 1 \bmod n. When n=91, there are 8 possible values for x, namely: 9, 16, 22, 29, 53, 74, 79, 81. Thus, C(91)=8. Find the sum of the positive numbers n \le 10^{11} for which C(n)=242. | 25,281 |
Consider equations of the form: a^2 + b^2 = N, 0 \le a \le b, a, b and N integer. For N=65 there are two solutions: a=1, b=8 and a=4, b=7. We call S(N) the sum of the values of a of all solutions of a^2 + b^2 = N, 0 \le a \le b, a, b and N integer. Thus S(65) = 1 + 4 = 5. Find \sum S(N), for all squarefree N only divis... | 25,282 |
For each integer p \gt 1 coprime to 10 there is a positive divisibility multiplier m \lt p which preserves divisibility by p for the following function on any positive integer, n: f(n) = (\text{all but the last digit of }n) + (\text{the last digit of }n) \cdot m. That is, if m is the divisibility multiplier for p, then... | 25,283 |
Let us define a balanced sculpture of order n as follows: A polyomino An arrangement of identical squares connected through shared edges; holes are allowed. made up of n + 1 tiles known as the blocks (n tiles) and the plinth (remaining tile); the plinth has its centre at position (x = 0, y = 0); the blocks have y-coord... | 25,284 |
Consider the triangles with integer sides a, b and c with a \le b \le c. An integer sided triangle (a,b,c) is called primitive if \gcd(a, b, c) \gcd(a,b,c)=\gcd(a,\gcd(b,c)) =1. How many primitive integer sided triangles exist with a perimeter not exceeding 10\,000\,000? | 25,285 |
A modified Collatz sequence of integers is obtained from a starting value a_1 in the following way: a_{n+1} = \, \,\, \frac {a_n} 3 \quad if a_n is divisible by 3. We shall denote this as a large downward step, "D". a_{n+1} = \frac {4 a_n+2} 3 \, \, if a_n divided by 3 gives a remainder of 1. We shall denote this as an... | 25,286 |
Given the values of integers 1 < a_1 < a_2 < \dots < a_n, consider the linear combination q_1 a_1+q_2 a_2 + \dots + q_n a_n=b, using only integer values q_k \ge 0. Note that for a given set of a_k, it may be that not all values of b are possible. For instance, if a_1=5 and a_2=7, there are no q_1 \ge 0 and q_2 \ge 0 su... | 25,287 |
How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed 10^8? | 25,288 |
A laborious ant walks randomly on a 5 \times 5 grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are 2, 3 or 4 possible moves at each step depending on the ant's position. At the start of the walk, a seed is placed on each sq... | 25,289 |
You are given a pizza (perfect circle) that has been cut into m \cdot n equal pieces and you want to have exactly one topping on each slice. Let f(m, n) denote the number of ways you can have toppings on the pizza with m different toppings (m \ge 2), using each topping on exactly n slices (n \ge 1). Reflections are con... | 25,290 |
\def\htmltext#1{\style{font-family:inherit;}{\text{#1}}} For non-negative integers m, n, the Ackermann function A(m,n) is defined as follows:
A(m,n) = \cases{
n+1 &\htmltext{ if }m=0\cr
A(m-1,1) &\htmltext{ if }m>0 \htmltext{ and } n=0\cr
A(m-1,A(m,n-1)) &\htmltext{ if }m>0 \htmltext{ and } n>0\cr
} For exam... | 25,291 |
Consider the triangle with sides 6, 8, and 10. It can be seen that the perimeter and the area are both equal to 24.
So the area/perimeter ratio is equal to 1. Consider also the triangle with sides 13, 14 and 15. The perimeter equals 42 while the area is equal to 84.
So for this triangle the area/perimeter ratio is eq... | 25,292 |
The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 376 2 = 141376. Let's call a number with this property a steady square. Steady squares can also be observed in other numbering systems. In the base 14 numbering system, th... | 25,293 |
Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform distribution. The square root of the sum (k \cdot a + 1)^2 + (k \cdot b + 1)^2 is then computed and rounded to the nearest integer. If the result is equal to k, he scores k points; otherwise he scores ... | 25,294 |
Barbara is a mathematician and a basketball player. She has found that the probability of scoring a point when shooting from a distance x is exactly (1 - x / q), where q is a real constant greater than 50. During each practice run, she takes shots from distances x = 1, x = 2, \dots, x = 50 and, according to her records... | 25,295 |
The quadtree encoding allows us to describe a 2^N \times 2^N black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this: the first bit deals with the complete 2^N \times 2^N region; "0" denotes a split: the current 2^n \times 2^n region is divided into 4 sub-regi... | 25,296 |
For any prime p the number N(p, q) is defined by
N(p, q) = \sum_{n = 0}^q T_n \cdot p^n with T_n generated by the following random number generator: S_0 = 290797 S_{n + 1} = S_n^2 \bmod 50515093 T_n = S_n \bmod p Let \operatorname{Nfac}(p, q) be the factorial of N(p, q). Let \operatorname{NF}(p, q) be the number of fac... | 25,297 |
Let C(x, y) be a circle passing through the points (x, y), (x, y + 1), (x + 1, y) and (x + 1, y + 1). For positive integers m and n, let E(m, n) be a configuration which consists of the m \cdot n circles: \{ C(x, y): 0 \le x \lt m, 0 \le y \lt n, x \text{ and } y \text{ are integers} \}. An Eulerian cycle on E(m, n) is... | 25,298 |
How many integers 0 \le n \lt 10^{18} have the property that the sum of the digits of n equals the sum of digits of 137n? | 25,299 |
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