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Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 133468. 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 a "bouncy" number; for example, 155349. As \(n\) increases, the proportion of..

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: \[\begin{split} \dfrac{1}{5} + \dfrac{1}{20} &= \dfrac{1}{4} \\ \dfrac{1}{6} + \dfrac{1}{12} &= \dfrac{1}{4} \\ \dfrac{1}{8} + \dfrac{1}{8} &= \dfrac{1}{4} \end{split}\] What is the least value of \(n\) for which the bumber..

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, it is possible to optimize the network by removing some edges and still e..

The Fibonacci sequence is defined by the recurrence relation: \({\rm F}_n = {\rm F}_{n-1} + {\rm F}_{n-2}, \) where \({\rm F}_1 =1 \) and \({\rm F}_2 = 1\) It turns out that \(\rm F_{541}\), 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 bot necessarily in order). And \(\rm F_{2749}\),..

If a box contains twenty-one colored 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, \(\rm P(BB)=\dfrac{15}{21} \times \dfrac{14}{20} = \dfrac{1}{2}\) The next such arrangement, for which there is exactly \(50%\) chance of taking two blue discs at random, is a box containing eighty-five blu..

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: \[ {\rm A}(-340,\; 495), \; {\rm B}(-153, \;-910), \; {\rm C}(835, \; -947)\] \[ {\rm X} (-175, \; 41),\; {\rm Y}(-421, \; -714),\; {\rm Z}(574, \; -645)\] It can be verified that triangle \(\rm ABC\) contains the origin..

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, 44 → 32 → 13 → 10 → 1 → 1 85 → 89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89 Therefore any chain that arrives at 1 or 89 will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at 1 or 89. How ma..

It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all. The square root of two is 1.41421356237309504880..., and the digital sum of the first one hundred decimal digits is 475. For the first one hundred natural numbers, find the total of the digital sums of ..

Let \(p(n)\) represent the number of different ways in which \(n\) coins can be separated into piles. For example, five coins can separated into piles in exactly seven different ways, so \(p(5)=7\). OOOOO OOOO O OOO OO OOO O O OO OO O OO O O O O O O O O Find the least value of \(n\) for which \(p(n)\) is divisible by one million. \(n\) 개의 동전을 각기 다른 방법으로 그룹 짓는 방법의 수를 \(p(n)\) 이라 하자. 예를 들어 5개의 동전에..

It is possible to write five as a sum in exactly six different ways: 4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 How many different ways can one hundred be written as a sum of at least two positive integers? 자연수 5를 두 자연수 이상의 합으로 나타내는 서로 다른 방법은 위와 같이 6가지가 있다. 자연수 100을 두 자연수 이상의 합으로 나타내는 서로 다른 방법은 몇 가지가 있을까? 이 문제를 풀기 위해서는 31번 문제를 잘 이해하고 와야 한다. 31번 문제를 이해했다면 76번은 31번과 유사한 문제로 변형..