2018-05-28 Basic Practice Problems Online

On this number pad, a nine-digit number is made by walking through each digit once, starting from a random digit and only moving to adjacent digits.

That number is then divided by its first digit. In this example, \[{ \color{orange}9}87456321 ÷ {\color{orange}9} = 109717369.\] Will this method always result in an integer?

True or False?

If \((a-b) \times (a-b) = (c-b) \times (c-b),\) then it must be true that \(a=c.\)

Bonus: If you think it's true, prove it; if you think it's false, find a counterexample.

A white square is inscribed in a blue circle, which is inscribed in a larger orange square.

Which area is larger, orange or blue?

Why is a seesaw always tilted to one side, while an empty beam balance is always balanced horizontally?

A. The center of mass of a seesaw is slightly above the hinge point, while that of a beam balance is slightly below the hinge point.

B. The center of mass of a seesaw is slightly below the hinge point, while that of a beam balance is slightly above the hinge point.

The diagram shows how identical equilateral triangles can be tiled to make a regular hexagon with no leftover tiles.

Which of the following numbers of triangles can also do this?

Regular hexagons can be made with 6 tiles, 24 tiles, and 54 tiles.

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 6 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.