Polygon Area
Let \(N\) be a positive composite integer. Let \(A\) be the area of the convex polygon whose vertices include all points \((x,y)\) such that \(x\) and \(y\) are both positive integers and \( y = x + \frac{N}{x} \).
| Part A | Given \(N=10\), what is \(A\)? | |
|---|---|---|
| Part B | Given \(N=12345678\), what is \(A\)? |
Five Consecutive Multiples
Let \(A,B,C,D,E\) be fixed positive constants. I'm thinking of the smallest positive integer \(n\) such that:
- \(n+1\) is divisible by \(A\)
- \(n+2\) is divisible by \(B\)
- \(n+3\) is divisible by \(C\)
- \(n+4\) is divisible by \(D\)
- \(n+5\) is divisible by \(E\)
| Part A | Given \(A=5\), \(B=7\), \(C=11\), \(D=13\) and \(E=17\), What is \(n\)? | |
|---|---|---|
| Part B | Given \(A=773\), \(B=787\), \(C=797\), \(D=809\) and \(E=811\), What is \(n\)? | |
| Part C | Given \(A=10^{12}+39\), \(B=10^{12}+61\), \(C=10^{12}+63\), \(D=10^{12}+91\) and \(E=10^{12}+121\), What is the remainder when \(n\) is divided by \(10^{12}+163\)? |