Problems

Ten people take part in a challenge. Each is given a hat, either black or white. Everyone can see the other hats, but not their own.

They speak one at a time. On your turn, you must say black or white. One special person may instead say a whole number between \(0\) and \(777\), and then also guess their colour.

Before the hats are placed, the group may agree on a strategy.

What is the largest number of people who can guarantee a correct guess?

What’s the smallest number of weights we need to weigh any number of grams from \(1\) to \(100\) in a balance scale, if your weights can be placed in any of the two plates of the scale?

Recall that when we write \(n!\) for some natural number \(n\), we mean \(1\times 2\times 3\times \cdots \times n\). You are given that \(20!=243290a0081766bc000,\) for some digits \(a,b,c\). Find those digits. You may want to recall the divisibility rule for \(9\): a number is divisible by \(9\) if and only if the sum of its digits is divisible by \(9\).

We have a square of side-length \(10\). A point \(P\) is drawn inside the square somewhere along the diagonal, and then the square is split and shaded as in the following diagram. Where should \(P\) be placed so that the shaded area is as smallest as possible? What will this shaded area be?

Let \(ABCD\) be a convex quadrilateral (convex means that all its internal angles are less than \(180^\circ\), i.e: it doesnt “bulge inwards"). Let \(E\) be the point of intersection of the diagonals. We are told that the triangles \(\triangle AED\) and \(\triangle BEC\) have equal areas. Show that \(AB\) and \(CD\) must be parallel.

For a natural number \(n\), we call the number \(1+2+3+\cdots + n\) the \(n^{\text{th}}\) triangular number, and we denote it by \(T_n\). Find \(T_n+T_{n-1}\) in terms of \(n\).

On a TV screen, each minute that passes, there is a number that flashes on the screen: first \(5\), then \(55\), then \(555\), and so on. Will any of these numbers be ever divisible by \(495\)? If so, which is the smallest?

For a natural number \(n\), define the \(n^{\text{th}}\) triangular number by \(T_n=1+2+3+\cdots+n\).

Show that \(3T_n+T_{n-1}\) is itself a triangular number, and determine which one.

Among \(12\) identical-looking balls, exactly one has a different weight (we do not know whether it is heavier or lighter than the others).

Using a balance scale, show how to determine the odd ball, and whether it is lighter or heavier, using only three weighings.

You are given \(68\) coins, and all of them have different weights. Using at most \(100\) weighings on a balance scale, find both the heaviest and the lightest coin.