Problems

Is \(\text{Nim}(2,5)\) a winning position or a losing position?

Let \(x,y\) be nonnegative integers. Determine when \(\text{Nim}(x,y)\) is a losing position and when it is a winning position.

Is \(\text{Nim}(1,2,3)\) a winning position or a losing position?

Is \(\text{Nim}(1,2,4,5,5)\) a winning position or a losing position?

When we write 137 in decimal, we mean \(1 \cdot 10^2 + 3 \cdot 10 + 7 \cdot 1\). If we write it instead using powers of 2, we have \(137 = 1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\). To tell apart binary representation from decimals, we can use the following notation: \(137 = (10001001)_2\).

What is the number 273 in binary? Note that using binary is useful for finding whether a particular Nim game is a winning position or a losing position.

Two fractions sum up to \(1\), but their difference is \(\frac1{10}\). What are they?

On her birthday, my grandma was asked how old she was. She said: "Start with the year I was born. Add the current year to it. Then, from the sum subtract the year I celebrated by \(20\)th birthday. From that, take away the year I was \(30\). The result will be \(16\)." How old is my grandma?

In the long addition above, each letter corresponds to a different digit. What is the sum \(D + O +G + C +A +T\)?

Let \(ABCDE\) be a regular pentagon. The point \(G\) is the midpoint of \(CD\), the point \(F\) is the midpoint of \(AE\). The lines \(EG\) and \(BF\) intersect at the point \(H\). Find the angle \(EHF\).

I have three positive integers. When you add them together, you get \(15\). When you multiply the three numbers together, you get \(120\).

What are the three numbers?