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The points P(x_1, y_1) and Q(x_2, y_2) are plotted at integer co-ordinates and are joined to the origin, O(0,0), to form \triangle OPQ. There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between 0 and 2 inclusive; that is, 0 \le x_1, y_1, x_2, y_2 \le 2. Given th... | 25,100 |
A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before. For example,
\begin{align}
&44 \to 32 \to 13 \to 10 \to \mathbf 1 \to \mathbf 1\\
&85 \to \mathbf{89} \to 145 \to 42 \to 20 \to 4 \to 16 \to 37 \to 58 \to \mathbf{89}
\end{align} The... | 25,101 |
By using each of the digits from the set, \{1, 2, 3, 4\}, exactly once, and making use of the four arithmetic operations (+, -, \times, /) and brackets/parentheses, it is possible to form different positive integer targets. For example, \begin{align}
8 &= (4 \times (1 + 3)) / 2\\
14 &= 4 \times (3 + 1 / 2)\\
19 &= 4 \t... | 25,102 |
It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle 5-5-6 has an area of 12 square units. We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than one uni... | 25,103 |
The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of 28 are 1, 2, 4, 7, and 14. As the sum of these divisors is equal to 28, we call it a perfect number. Interestingly the sum of the proper divisors of 220 is 284 and the sum of the proper divisors of 284 ... | 25,104 |
Su Doku (Japanese meaning number place ) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (or zeros)... | 25,105 |
The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 2^{6972593} - 1; it contains exactly 2\,098\,960 digits. Subsequently other Mersenne primes, of the form 2^p - 1, have been found which contain more digits. However, in 2004 there was found a massive non... | 25,106 |
By replacing each of the letters in the word CARE with 1, 2, 9, and 6 respectively, we form a square number: 1296 = 36^2. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: 9216 = 96^2. We shall call CARE (and RACE) a square anagram word pair and specify ... | 25,107 |
Comparing two numbers written in index form like 2^{11} and 3^7 is not difficult, as any calculator would confirm that 2^{11} = 2048 \lt 3^7 = 2187. However, confirming that 632382^{518061} \gt 519432^{525806} would be much more difficult, as both numbers contain over three million digits. Using base_exp.txt (right cli... | 25,108 |
If a box contains twenty-one coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, P(\text{BB}) = (15/21) \times (14/20) = 1/2. The next such arrangement, for which there is exactly 50\% chance of taking two bl... | 25,109 |
If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence. As an example, let us consider the sequence of cube numbers. This is defined by the generating function, u_n = n^3: 1... | 25,110 |
Three distinct points are plotted at random on a Cartesian plane, for which -1000 \le x, y \le 1000, such that a triangle is formed. Consider the following two triangles: \begin{gather}
A(-340,495), B(-153,-910), C(835,-947)\\
X(-175,41), Y(-421,-714), Z(574,-645)
\end{gather} It can be verified that triangle ABC conta... | 25,111 |
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true: S(B) \ne S(C); that is, sums of subsets cannot be equal. If B contains more elements than C then S(B) \gt S(C). If S(A) is minimised for a ... | 25,112 |
The Fibonacci sequence is defined by the recurrence relation: F_n = F_{n - 1} + F_{n - 2}, where F_1 = 1 and F_2 = 1. It turns out that F_{541}, which contains 113 digits, is the first Fibonacci number for which the last nine digits are 1-9 pandigital (contain all the digits 1 to 9, but not necessarily in order). And F... | 25,113 |
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true: S(B) \ne S(C); that is, sums of subsets cannot be equal. If B contains more elements than C then S(B) \gt S(C). For example, \{81, 88, 75, ... | 25,114 |
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true: S(B) \ne S(C); that is, sums of subsets cannot be equal. If B contains more elements than C then S(B) \gt S(C). For this problem we shall a... | 25,115 |
The following undirected network consists of seven vertices and twelve edges with a total weight of 243. The same network can be represented by the matrix below. A B C D E F G A - 16 12 21 - - - B 16 - - 17 20 - - C 12 - - 28 - 31 - D 21 17 28 - 18 19 23 E - 20 - 18 - - 11 F - - 31 19 - - 27 G - - - 23 11 27 - However,... | 25,116 |
In the following equation x, y, and n are positive integers. \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n} For n = 4 there are exactly three distinct solutions: \begin{align}
\dfrac{1}{5} + \dfrac{1}{20} &= \dfrac{1}{4}\\
\dfrac{1}{6} + \dfrac{1}{12} &= \dfrac{1}{4}\\
\dfrac{1}{8} + \dfrac{1}{8} &= \dfrac{1}{4}
\end{align... | 25,117 |
In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty. The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions inside... | 25,118 |
In the following equation x, y, and n are positive integers. \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n} It can be verified that when n = 1260 there are 113 distinct solutions and this is the least value of n for which the total number of distinct solutions exceeds one hundred. What is the least value of n for which the... | 25,119 |
Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by 11, 2222 is divisible by 22, and so on. But there are nine 4-digit primes containing three ones:
1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111. We shall say that M(n, d) represents the maximum... | 25,120 |
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468. Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420. We shall call a positive integer that is neither increasing nor decreasing... | 25,121 |
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468. Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420. We shall call a positive integer that is neither increasing nor decreasing... | 25,122 |
A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one grey square. There are exactly seventeen ways of doing this. How many ways can a row measuring fifty units in len... | 25,123 |
NOTE: This is a more difficult version of Problem 114 . A row measuring n units in length has red blocks with a minimum length of m units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square. Let the fill-count function, F(m, n), represent the... | 25,124 |
A row of five grey square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (length two), green (length three), or blue (length four). If red tiles are chosen there are exactly seven ways this can be done. If green tiles are chosen there are three ways. And if blue tiles are cho... | 25,125 |
Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (measuring three units), and blue tiles (measuring four units), it is possible to tile a row measuring five units in length in exactly fifteen different ways. How many ways can a row measuring fifty units... | 25,126 |
Using all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set \{2,5,47,89,631\}, all of the elements belonging to it are prime. How many distinct sets containing each of the digits one through nine exactly once contain only prime ele... | 25,127 |
The number 512 is interesting because it is equal to the sum of its digits raised to some power: 5 + 1 + 2 = 8, and 8^3 = 512. Another example of a number with this property is 614656 = 28^4. We shall define a_n to be the nth term of this sequence and insist that a number must contain at least two digits to have a sum.... | 25,128 |
Let r be the remainder when (a - 1)^n + (a + 1)^n is divided by a^2. For example, if a = 7 and n = 3, then r = 42: 6^3 + 8^3 = 728 \equiv 42 \mod 49. And as n varies, so too will r, but for a = 7 it turns out that r_{\mathrm{max}} = 42. For 3 \le a \le 1000, find \sum r_{\mathrm{max}}. | 25,129 |
A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted. After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random. The player pays £1 to play and wins if they have taken more blue discs than red di... | 25,130 |
The most naive way of computing n^{15} requires fourteen multiplications:
n \times n \times \cdots \times n = n^{15}. But using a "binary" method you can compute it in six multiplications: \begin{align}
n \times n &= n^2\\
n^2 \times n^2 &= n^4\\
n^4 \times n^4 &= n^8\\
n^8 \times n^4 &= n^{12}\\
n^{12} \times n^2 &= n... | 25,131 |
Let p_n be the nth prime: 2, 3, 5, 7, 11, \dots, and let r be the remainder when (p_n - 1)^n + (p_n + 1)^n is divided by p_n^2. For example, when n = 3, p_3 = 5, and 4^3 + 6^3 = 280 \equiv 5 \mod 25. The least value of n for which the remainder first exceeds 10^9 is 7037. Find the least value of n for which the remaind... | 25,132 |
The radical of n, \operatorname{rad}(n), is the product of the distinct prime factors of n. For example, 504 = 2^3 \times 3^2 \times 7, so \operatorname{rad}(504) = 2 \times 3 \times 7 = 42. If we calculate \operatorname{rad}(n) for 1 \le n \le 10, then sort them on \operatorname{rad}(n), and sorting on n if the radica... | 25,133 |
The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2. There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is 4164. Note that 1 = 0^2 + 1^2 has... | 25,134 |
The minimum number of cubes to cover every visible face on a cuboid measuring 3 \times 2 \times 1 is twenty-two. If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one-hundred and... | 25,135 |
The radical of n, \operatorname{rad}(n), is the product of distinct prime factors of n. For example, 504 = 2^3 \times 3^2 \times 7, so \operatorname{rad}(504) = 2 \times 3 \times 7 = 42. We shall define the triplet of positive integers (a, b, c) to be an abc-hit if: \gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1 a \lt b a + ... | 25,136 |
A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering the tiles 2 to 7 in an anti-clockwise direction. New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first thr... | 25,137 |
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111. Given that n is a positive integer and \gcd(n, 10) = 1, it can be shown that there always exists a value, k, for which R(k) is divisible by n, and let A(n) be the least such value of k... | 25,138 |
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111. Given that n is a positive integer and \gcd(n, 10) = 1, it can be shown that there always exists a value, k, for which R(k) is divisible by n, and let A(n) be the least such value of k... | 25,139 |
There are some prime values, p, for which there exists a positive integer, n, such that the expression n^3 + n^2p is a perfect cube. For example, when p = 19, 8^3 + 8^2 \times 19 = 12^3. What is perhaps most surprising is that for each prime with this property the value of n is unique, and there are only four such prim... | 25,140 |
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k. For example, R(10) = 1111111111 = 11 \times 41 \times 271 \times 9091, and the sum of these prime factors is 9414. Find the sum of the first forty prime factors of R(10^9). | 25,141 |
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111. Let us consider repunits of the form R(10^n). Although R(10), R(100), or R(1000) are not divisible by 17, R(10000) is divisible by 17. Yet there is no value of n for which R(10^n) will... | 25,142 |
Consider the consecutive primes p_1 = 19 and p_2 = 23. It can be verified that 1219 is the smallest number such that the last digits are formed by p_1 whilst also being divisible by p_2. In fact, with the exception of p_1 = 3 and p_2 = 5, for every pair of consecutive primes, p_2 \gt p_1, there exist values of n for wh... | 25,143 |
Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the positive integer, n, for which the equation, x^2 - y^2 - z^2 = n, has exactly two solutions is n = 27:
34^2 - 27^2 - 20^2 = 12^2 - 9^2 - 6^2 = 27. It turns out that n = 1155 is the least value which has ... | 25,144 |
The positive integers, x, y, and z, are consecutive terms of an arithmetic progression. Given that n is a positive integer, the equation, x^2 - y^2 - z^2 = n, has exactly one solution when n = 20:
13^2 - 10^2 - 7^2 = 20. In fact there are twenty-five values of n below one hundred for which the equation has a unique sol... | 25,145 |
Consider the infinite polynomial series A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots, where F_k is the kth term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, \dots; that is, F_k = F_{k-1} + F_{k-2}, F_1 = 1 and F_2 = 1. For this problem we shall be interested in values of x for which A_F(x) is a positive integer. Surprisi... | 25,146 |
Consider the isosceles triangle with base length, b = 16, and legs, L = 17. By using the Pythagorean theorem it can be seen that the height of the triangle, h = \sqrt{17^2 - 8^2} = 15, which is one less than the base length. With b = 272 and L = 305, we get h = 273, which is one more than the base length, and this is t... | 25,147 |
Let (a, b, c) represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to form a square with length c. For example, (3, 4, 5) triangles can be placed together to form a 5 by 5 square with a 1 by 1 hole in the middle and it can be seen that the ... | 25,148 |
Consider the infinite polynomial series A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots, where G_k is the kth term of the second order recurrence relation G_k = G_{k-1} + G_{k-2}, G_1 = 1 and G_2 = 4; that is, 1, 4, 5, 9, 14, 23, \dots. For this problem we shall be concerned with values of x for which A_G(x) is a positive ... | 25,149 |
A positive integer, n, is divided by d and the quotient and remainder are q and r respectively. In addition d, q, and r are consecutive positive integer terms in a geometric sequence, but not necessarily in that order. For example, 58 divided by 6 has quotient 9 and remainder 4. It can also be seen that 4, 6, 9 are con... | 25,150 |
Find the smallest x + y + z with integers x \gt y \gt z \gt 0 such that x + y, x - y, x + z, x - z, y + z, y - z are all perfect squares. | 25,151 |
Let ABC be a triangle with all interior angles being less than 120 degrees. Let X be any point inside the triangle and let XA = p, XC = q, and XB = r. Fermat challenged Torricelli to find the position of X such that p + q + r was minimised. Torricelli was able to prove that if equilateral triangles AOB, BNC and AMC are... | 25,152 |
In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around on the mirrors, and eventually works its way back out. The specific white cell we will be considering is an ellipse with the equation 4x^2 + y^2 = 100. The section corresponding to ... | 25,153 |
Some positive integers n have the property that the sum [n + \operatorname{reverse}(n)] consists entirely of odd (decimal) digits. For instance, 36 + 63 = 99 and 409 + 904 = 1313. We will call such numbers reversible ; so 36, 63, 409, and 904 are reversible. Leading zeroes are not allowed in either n or \operatorname{r... | 25,154 |
The smallest positive integer n for which the numbers n^2 + 1, n^2 + 3, n^2 + 7, n^2 + 9, n^2 + 13, and n^2 + 27 are consecutive primes is 10. The sum of all such integers n below one-million is 1242490. What is the sum of all such integers n below 150 million? | 25,155 |
In a 3 \times 2 cross-hatched grid, a total of 37 different rectangles could be situated within that grid as indicated in the sketch. There are 5 grids smaller than 3 \times 2, vertical and horizontal dimensions being important, i.e. 1 \times 1, 2 \times 1, 3 \times 1, 1 \times 2 and 2 \times 2. If each of them is cros... | 25,156 |
We can easily verify that none of the entries in the first seven rows of Pascal's triangle are divisible by 7: 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 However, if we check the first one hundred rows, we will find that only 2361 of the 5050 entries are not divisible by 7. Find the number of entries ... | 25,157 |
Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or anti-diagonal) is 16 (= 8 + 7 + 1). -2 5 3 2 9 -6 5 1 3 2 7 3 -1 8 -4 8 Now, let us repeat the search, but on a much larger scale: First, generate four million pseudo-ra... | 25,158 |
In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest possible. In the example below, it can be easily verified that the marked triangle satisfies this condition having a sum of −42. We wish to make such a triangular array wi... | 25,159 |
A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of size A5. Every Monday morning, the supervisor opens a new envelope, containing a large sheet of the special paper with size A1. The supervisor proceeds to cut it in half, thus getting two sheets of size... | 25,160 |
There are several ways to write the number \dfrac{1}{2} as a sum of square reciprocals using distinct integers. For instance, the numbers \{2,3,4,5,7,12,15,20,28,35\} can be used: \begin{align}\dfrac{1}{2} &= \dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2} + \dfrac{1}{5^2} +\\
&\quad \dfrac{1}{7^2} + \dfrac{1}{12^2} +... | 25,161 |
As we all know the equation x^2=-1 has no solutions for real x. If we however introduce the imaginary number i this equation has two solutions: x=i and x=-i. If we go a step further the equation (x-3)^2=-4 has two complex solutions: x=3+2i and x=3-2i. x=3+2i and x=3-2i are called each others' complex conjugate. Numbers... | 25,162 |
A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level. Then, we calculate the number of paths leading from the apex to each position: A path starts at the apex and progresses downwards to any of the three spheres directly below the current posit... | 25,163 |
An electric circuit uses exclusively identical capacitors of the same value C. The capacitors can be connected in series or in parallel to form sub-units, which can then be connected in series or in parallel with other capacitors or other sub-units to form larger sub-units, and so on up to a final circuit. Using this s... | 25,164 |
Starting from zero the natural numbers are written down in base 10 like this: 0\,1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\cdots Consider the digit d=1. After we write down each number n, we will update the number of ones that have occurred and call this number f(n,1). The first values for f(n,1), then, are as follows: \be... | 25,165 |
Consider the diophantine equation \frac 1 a + \frac 1 b = \frac p {10^n} with a, b, p, n positive integers and a \le b. For n=1 this equation has 20 solutions that are listed below:
\begin{matrix}
\frac 1 1 + \frac 1 1 = \frac{20}{10} & \frac 1 1 + \frac 1 2 = \frac{15}{10} & \frac 1 1 + \frac 1 5 = \frac{12}{10} & \fr... | 25,166 |
Taking three different letters from the 26 letters of the alphabet, character strings of length three can be formed. Examples are 'abc', 'hat' and 'zyx'. When we study these three examples we see that for 'abc' two characters come lexicographically after its neighbour to the left. For 'hat' there is exactly one charact... | 25,167 |
A composite number can be factored many different ways.
For instance, not including multiplication by one, 24 can be factored in 7 distinct ways: \begin{align}
24 &= 2 \times 2 \times 2 \times 3\\
24 &= 2 \times 3 \times 4\\
24 &= 2 \times 2 \times 6\\
24 &= 4 \times 6\\
24 &= 3 \times 8\\
24 &= 2 \times 12\\
24 &= 2... | 25,168 |
For any N, let f(N) be the last five digits before the trailing zeroes in N!. For example, 9! = 362880 so f(9)=36288 10! = 3628800 so f(10)=36288 20! = 2432902008176640000 so f(20)=17664 Find f(1\,000\,000\,000\,000). | 25,169 |
A triomino is a shape consisting of three squares joined via the edges.
There are two basic forms: If all possible orientations are taken into account there are six: Any n by m grid for which n \times m is divisible by 3 can be tiled with triominoes. If we consider tilings that can be obtained by reflection or rotation... | 25,170 |
In the hexadecimal number system numbers are represented using 16 different digits:
0,1,2,3,4,5,6,7,8,9,\mathrm A,\mathrm B,\mathrm C,\mathrm D,\mathrm E,\mathrm F. The hexadecimal number \mathrm{AF} when written in the decimal number system equals 10 \times 16 + 15 = 175. In the 3-digit hexadecimal numbers 10\mathrm A... | 25,171 |
Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the size 1 triangle in the sketch below. Sixteen triangles of either different shape or size or orientation or location can now be observed in that triangle. Using size 1 triangles as build... | 25,172 |
How many 20 digit numbers n (without any leading zero) exist such that no three consecutive digits of n have a sum greater than 9? | 25,173 |
A segment is uniquely defined by its two endpoints. By considering two line segments in plane geometry there are three possibilities: the segments have zero points, one point, or infinitely many points in common. Moreover when two segments have exactly one point in common it might be the case that that common point is ... | 25,174 |
A 4 \times 4 grid is filled with digits d, 0 \le d \le 9. It can be seen that in the grid
\begin{matrix}
6 & 3 & 3 & 0\\
5 & 0 & 4 & 3\\
0 & 7 & 1 & 4\\
1 & 2 & 4 & 5
\end{matrix}
the sum of each row and each column has the value 12. Moreover the sum of each diagonal is also 12. In how many ways can you fill a 4 \times... | 25,175 |
For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)_1 = a, U(a,b)_2 = b and for k \gt 2,
U(a,b)_k is the smallest integer greater than U(a,b)_{k - 1} which can be written in exactly one way as the sum of two distinct previous members of U(a,b). For example, the sequence U(1,2) begins with 1... | 25,176 |
Consider the number 142857. We can right-rotate this number by moving the last digit (7) to the front of it, giving us 714285. It can be verified that 714285 = 5 \times 142857. This demonstrates an unusual property of 142857: it is a divisor of its right-rotation. Find the last 5 digits of the sum of all integers n, 10... | 25,177 |
Define f(0)=1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using each power no more than twice. For example, f(10)=5 since there are five different ways to express 10: \begin{align}
& 1 + 1 + 8\\
& 1 + 1 + 4 + 4\\
& 1 + 1 + 2 + 2 + 4\\
& 2 + 4 + 4\\
& 2 + 8
\end{align} ... | 25,178 |
Take the number 6 and multiply it by each of 1273 and 9854: \begin{align}
6 \times 1273 &= 7638\\
6 \times 9854 &= 59124
\end{align} By concatenating these products we get the 1 to 9 pandigital 763859124. We will call 763859124 the "concatenated product of 6 and (1273,9854)". Notice too, that the concatenation of the i... | 25,179 |
For a positive integer n, let f(n) be the sum of the squares of the digits (in base 10) of n, e.g. \begin{align}
f(3) &= 3^2 = 9,\\
f(25) &= 2^2 + 5^2 = 4 + 25 = 29,\\
f(442) &= 4^2 + 4^2 + 2^2 = 16 + 16 + 4 = 36\\
\end{align} Find the last nine digits of the sum of all n, 0 \lt n \lt 10^{20}, such that f(n) is a perfe... | 25,180 |
How many 18-digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n? | 25,181 |
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. For example, using exactly thirty-two square tiles we can form two different square laminae: With one-hundred tiles, and not necessarily using all of the tiles at one time, it is pos... | 25,182 |
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. Given eight tiles it is possible to form a lamina in only one way: 3 \times 3 square with a 1 \times 1 hole in the middle. However, using thirty-two tiles it is possible to form two ... | 25,183 |
Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice. For example, f(10)=5 since there are five different ways to express 10: 10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1. It can be shown that for every fraction p / q (p \gt 0, q \gt 0) there exists ... | 25,184 |
The four right-angled triangles with sides (9,12,15), (12,16,20), (5,12,13) and (12,35,37) all have one of the shorter sides (catheti) equal to 12. It can be shown that no other integer sided right-angled triangle exists with one of the catheti equal to 12. Find the smallest integer that can be the length of a cathetus... | 25,185 |
Let ABCD be a convex quadrilateral, with diagonals AC and BD. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles. For example, at vertex A, the two angles are CAD, CAB. We call such a quadrilateral for which all eight corner angles have integer values when measured in de... | 25,186 |
Consider the number 45656. It can be seen that each pair of consecutive digits of 45656 has a difference of one. A number for which every pair of consecutive digits has a difference of one is called a step number. A pandigital number contains every decimal digit from 0 to 9 at least once. How many pandigital step numb... | 25,187 |
Find the number of integers 1 \lt n \lt 10^7, for which n and n + 1 have the same number of positive divisors. For example, 14 has the positive divisors 1, 2, 7, 14 while 15 has 1, 3, 5, 15. | 25,188 |
For any integer n, consider the three functions \begin{align}
f_{1, n}(x, y, z) &= x^{n + 1} + y^{n + 1} - z^{n + 1}\\
f_{2, n}(x, y, z) &= (xy + yz + zx) \cdot (x^{n - 1} + y^{n - 1} - z^{n - 1})\\
f_{3, n}(x, y, z) &= xyz \cdot (x^{n - 2} + y^{n - 2} - z^{n - 2})
\end{align} and their combination
f_n(x, y, z) = f_{1,... | 25,189 |
Having three black objects B and one white object W they can be grouped in 7 ways like this: (BBBW) (B,BBW) (B,B,BW) (B,B,B,W) (B,BB,W) (BBB,W) (BB,BW) In how many ways can sixty black objects B and forty white objects W be thus grouped? | 25,190 |
The RSA encryption is based on the following procedure: Generate two distinct primes p and q. Compute n = pq and \phi = (p - 1)(q - 1). Find an integer e, 1 \lt e \lt \phi, such that \gcd(e, \phi) = 1. A message in this system is a number in the interval [0, n - 1]. A text to be encrypted is then somehow converted to m... | 25,191 |
Let N be a positive integer and let N be split into k equal parts, r = N/k, so that N = r + r + \cdots + r. Let P be the product of these parts, P = r \times r \times \cdots \times r = r^k. For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.2^5 = 51.53632. Let M(N) = P_{\mat... | 25,192 |
Consider the set I_r of points (x,y) with integer co-ordinates in the interior of the circle with radius r, centered at the origin, i.e. x^2 + y^2 \lt r^2. For a radius of 2, I_2 contains the nine points (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1) and (1,-1). There are eight triangles having all three v... | 25,193 |
The game Number Mind is a variant of the well known game Master Mind. Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you'd be told that you have one correct... | 25,194 |
Here are the records from a busy telephone system with one million users: RecNr Caller Called 1 200007 100053 2 600183 500439 3 600863 701497 \cdots \cdots \cdots The telephone number of the caller and the called number in record n are \operatorname{Caller}(n) = S_{2n-1} and \operatorname{Called}(n) = S_{2n} where S_{1... | 25,195 |
A composite is a number containing at least two prime factors. For example, 15 = 3 \times 5; 9 = 3 \times 3; 12 = 2 \times 2 \times 3. There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors:
4, 6, 9, 10, 14, 15, 21, 22, 25, 26. How many composite integers, n \lt 10^8, ha... | 25,196 |
The hyperexponentiation or tetration of a number a by a positive integer b, denoted by a\mathbin{\uparrow \uparrow}b or ^b a, is recursively defined by: a \mathbin{\uparrow \uparrow} 1 = a, a \mathbin{\uparrow \uparrow} (k+1) = a^{(a \mathbin{\uparrow \uparrow} k)}. Thus we have e.g. 3 \mathbin{\uparrow \uparrow} 2 = 3... | 25,197 |
Consider the following configuration of 64 triangles: We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same colour. Such a colouring shall be called valid. Here, two triangles are said to be neighbouring if they share an edge. ... | 25,198 |
Let S_m = (x_1, x_2, \dots , x_m) be the m-tuple of positive real numbers with x_1 + x_2 + \cdots + x_m = m for which P_m = x_1 \cdot x_2^2 \cdot \cdots \cdot x_m^m is maximised. For example, it can be verified that \lfloor P_{10}\rfloor = 4112 (\lfloor \, \rfloor is the integer part function). Find \sum\limits_{m = 2}... | 25,199 |
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