Problems

What’s bigger out of \(99!\) and \(50^{99}\)?

Let \(x\) be the sum of digits of \(4444^{4444}\). Let \(y\) be the sum of digits of \(x\). What’s the sum of the digits of \(y\)?

Using the fact that \(\log_{10}(3)\approx0.4771\), \(\log_{10}(5)\approx0.698\) and \(\log_{10}(6)\approx0.778\) all correct to three or four decimal places (check), show that \(5\times10^{47}<3^{100}<6\times10^{47}\). How many digits does \(3^{100}\) have, and what’s its first digit?

What is the least \(N\) such that \(\sum_{n=1}^N1/n\ge100\)?

Evaluate \(a(4,4)\) for the function \(a(m,n)\), which is defined for integers \(m,n\ge0\) by \[\begin{align*} a(0,n)&=n+1\text{, if }n\ge0;\\ a(m,0)&=a(m-1,1)\text{, if }m>0;\\ a(m,n)&=a(m-1,a(m,n-1))\text{, if }m>0\text{, and }n>0. \end{align*}\]

Let \(s\) and \(t\) be two positive integers. Can we have \(s^2=t^2+2\)?

What is the following as a single fraction? \[\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{98\times99}+\frac{1}{99\times100}.\]

Are there any two-digit numbers which are the product of their digits?

Find, with proof, all integer solutions of \(a^3+b^3=9\).

Suppose that \((x_1,y_1),(x_2,y_2)\) are solutions to Pell’s equation \(x^2-dy^2 = 1\). Show that \((x_1x_2+dy_1y_2,x_1y_2+x_2y_2)\) also satisfies the same equation.