코딩 연습

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 \(r=42: \; 6^3 +8^3 = 728 \equiv 42 \; \rm mod\; 49\). And as \(n\) varies, so too will \(r\), but for \(a=7\) it turns out that \(r_{\rm max} =42\). For \(3 \le a \le 1000\), find \(\sum r_{\rm max}\). \((a-1)^n + (a+1)^n\) 을 \(a^2\) 으로 나누었을 때의 나머지를 \(r\) 라고 하자. 예를 들어, \(a=7\) 이..

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 elements? 1 부터 9까지 모든 자연수들을 한 번씩만 사용하여 자연수들을 만든다면 여러 가지 가능성이 있다. 예를 들어, 집합 {2, ..

Using a combination of black 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 in length be tiled? 길이 1인 정사각형 모양의 검은색 타일과 길이 2인 직사각형 모양의 빨간색 타일, 길이 3인 직사각형 모양의 초..

A row of five black square tiles is to have a number of its tiles replaced with colored 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 of blue tiles are chosen there are two ways. Assuming that colors cannot be mixed there are 7+3+2=1..

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..