We solve problems

Problems Discrete Mathematics Set theory and logic Mathematical logic Puzzles

Explain why you can’t rotate the sides on a normal Rubik’s cube to get to the following picture (with no removing stickers, painting, or other cheating allowed).

Problem #WSP-000494

Problems Number Theory Numeral systems Decimal number system Algebra Algebraic equations and systems of equations Diophantine equations

10–16 2.0

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

Problem #WSP-000495

Problems Arithmetic puzzles

10–13 2.0

The sum of Matt’s and Parker’s ages is \(63\) years. Matt is twice as old as Parker was when Matt was as old as Parker is now. How old are they? (Show that there’s no other ages that they could have)

Problem #WSP-000496

Problems Discrete Mathematics Combinatorics

11–14 2.0

How many \(10\)-digit numbers are there such that the sum of their digits is \(3\)?

Problem #WSP-000497

Problems Discrete Mathematics Combinatorics

11–14 3.0

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 \(n\)?

Problem #WSP-000498

Problems Geometry Plane geometry Triangles

10–16 2.0

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.

Problem #WSP-000499

Problems Algebra Algebraic equations and systems of equations Diophantine equations

10–16 3.0

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

Problem #WSP-000500

Problems Methods Mathematical induction

2.0

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?

Problem #WSP-000501

Problems Methods Mathematical induction

2.0

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?

Problem #WSP-000502

Problems Methods Mathematical induction

2.0

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?