코딩 연습

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 \(-42\). We wish to make such a triangular array with one thousand rows, so we generate \(500500\) pseudo-random numbers \(s_k\)..

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 \(16\; (=8+7+1)\). −25329−6513273−18−4 8 Now, let us repeat the search, but on a much larger scale: First, generate four million pseudo-random numbers using a specific form of what is known as a "Lagged Fibonacci Generator": For ..

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 which are not divisible by \(7\) in the first one billion \(\l..

In a\(3 \times 2\) cross-hatched grid, a total of \(37\) different rectangles could be situated withing 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 cross-hatched, the following number of differe..

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? \(n^2 +1\), \(n^2 +3\), \(n^2+7\), \(n^2+9\), \(n^2+13\), \(n^2+27\) 이 연속적인 소수가 되는 최소의 자연수 \(n\) 은..

Some positive integers \(n\) have the property that the sum \([n+{\rm reverse}(n)]\) 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 \({\rm reverse}(n)\). There are \(120\) reversible numbers below on-thousand. H..

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 \(4^x+y^2=100\). The section corresponding to \(-0.01 \le x \le 0.01\) at the top is missing, allowing the light to enter and..

Let \(\rm ABC\) be a triangle with all interior angles being less than \(120\) degrees. Let \(\rm X\) be any point inside the triangle and let \({\rm XA}=p,\; {\rm XC}=q\), and \({\rm XB}=r\). Fermat challenged Torricelli to find the position of \(\rm X\) such that \(p+q+r\) was minimized. Torricelli was able to prove that if equilateral triangles \(\rm AOB, \; BNC\) and \(\rm AMC\) are construc..

Find the smallest \(x+y+z\) with integers \(x>y>z>0\) such that \(x+y\), \(x-y\), \(x+z\), \(x-z\), \(y+z\), \(y-z\) are all perfect squares. \(x>y>z>0\) 을 만족하는 자연수 \(x, \; y\; z\) 에 대해서 \(x+y\), \(x-y\), \(x+z\), \(x-z\), \(y+z\), \(y-z\) 가 모두 완전제곱수가 되는 \(x+y+z\) 의 최솟값을 구하시오. 늘 그렇듯이 처음엔 무식하게 하나하나 완전제곱수인 것을 모두 확인하는 방법을 사용했지만, 아니나 다를까 시간이 너무 오래 걸린다. 그래서 다음의 방법을 생각 !! \[x+y=a^2 \label{a}\tag{1}\] ..

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 consecutive terms in a geometric ..