Problems

The sum of digits of a positive integer \(n\) is the same as the number of digits of \(n\). What are the possible products of the digits of \(n\)?

In the triangle \(\triangle ABC\), the angle \(\angle ACB=60^{\circ}\), marked at the top. The angle bisectors \(AD\) and \(BE\) intersect at the point \(I\).

Find the angle \(\angle AIB\), marked in red.

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

Alice and Bob were playing outdoors. A mean lady told them that at least one of them has a muddy face and everyone who has a muddy face must step forward at the same time on the count of three. Then the mean lady will leave them alone.

If a child with clean face steps forward, he is punished. If nobody steps forward, then the mean lady will do the count again. The children are not allowed to signal to each other. How can Alice and Bob avoid punishment?

Alice, Bob and Claire were playing outdoors. A mean lady told them that at least one of them has a muddy face and everyone who has a muddy face must step forward at the same time on the count of three. Then the mean lady will leave them alone.

If a child with clean face steps forward, he is punished. If nobody steps forward, then the mean lady will do the count again. The children are not allowed to signal to each other. How can Alice, Bob and Claire avoid punishment?

Alice, Bob, Claire and Daniel were playing outdoors. A mean lady told them that at least one of them has a muddy face and everyone who has a muddy face must step forward at the same time on the count of three. Then the mean lady will leave them alone.

If a child with clean face steps forward, he is punished. If nobody steps forward, then the mean lady will do the count again. The children are not allowed to signal to each other. How can Alice, Bob, Claire and Daniel avoid punishment?

A group of children were playing outdoors. A mean lady told them that at least one of them has a muddy face and everyone who has a muddy face must step forward at the same time on the count of three. Then the mean lady will leave them alone.

If a child with clean face steps forward, he is punished. If nobody steps forward, then the mean lady will do the count again. The children are not allowed to signal to each other. How can they avoid punishment?

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.

Suppose that \(x+y\sqrt{d}>1\) gives a solution to Pell’s equation. Show that \(x\geq 2\) and \(y\geq 1\). Can the bounds be achieved?

Every year the citizens of the planet “Lotsofteeth" enter a contest to see who has the most teeth.
This year the judge notices:

1. Nobody has 0 teeth (everyone has at least 1).

2. There are more people in the contest than the most teeth that any one person has. (For example, if the most teeth anyone has is 27, then there are more than 27 people participating in the contest.)

Must there be two people who have exactly the same number of teeth? Explain why.