Almost Square Triangles
A perfect square is any integer in the form \(n^2\) where \(n\) is an integer. A triangle number is any integer in the form \(\frac{n(n+1)}{2}\) where \(n\) is an integer. An almost-square-triangle is any triangle number which is one less than a perfect square. For example: \(3\) is an almost-square-triangle because \(3\) is a triangle number and \(4\) is a perfect square. Let \(S(x)\) denote the sum of all almost-square-triangles less than \(x\). Since \(3\) and \(0\) are the only almost-square-triangles less than \(10\), we have \(S(10) = 3\).| Part A | What is \(S(1000)\)? | |
|---|---|---|
| Part B | What is the remainder when \(S(10^{12})\) is divided by \((10^9+7)\)? | |
| Part C | What is the remainder when \(S(10^{1000})\) is divided by \((10^9+7)\)? |
Four Primes
There are four unknown prime numbers \(a,b,c,d\). You are given the values of the products: \(ab\) and \(bc\) and \(cd\). Your task is to determine the value of \(a+b+c+d\). For example, if \(ab=14\) and \(bc=6\) and \(cd=33\) then the only solution is \(a=7, b=2, c=3, d=11\) and so the answer would be \(a+b+c+d=7+2+3+11=23\).| Part A | \(ab=\) 1679, \(bc=\) 437, \(cd=\) 2147. What is \(a+b+c+d\)? | |
|---|---|---|
| Part B | \(ab=\) 289626105619, \(bc=\) 429039650437, \(cd=\) 134197842017. | |
| Part C | \(ab=\) 9924230848537907113490232026 \(bc=\) 9569803352783866555284669449 \(cd=\) 9940732613767440074616546364 |