Mathcounts

Each morning, Khalid and Jack arrive at the dining hall separately, at a random time from 8:00 to 9:00 a.m., and each remains there for exactly 15 minutes while eating. What is the probability they will see each other in the dining hall tomorrow morning? Express your answer as a common fraction. 

The line with equation ax + by = c, where a, b and c are positive, forms a right triangle with legs on the x- and y-axes. What is the area of the triangle? Express your answer as a common fraction in terms of a, b and c.

If a, b, c and d represent four different positive integers such that \(a^2+d^2=b^2+c^2\), what is the least possible value of |d − a|?

To weigh an object by using a balance scale, Brady places the object on one side of the scale and places enough weights on each side to make the two sides of the scale balanced. Brady’s set of weights contains the minimum number necessary to measure the whole-number weight of any object from 1 to 40 pounds, inclusive. What is the greatest weight, in pounds, of a weight in Brady’s set?

Daniel began painting a room at 9:00 a.m. Yeong, who can paint twice as fast as Daniel, started helping Daniel at 9:20 a.m., and they worked together until the room was fully painted at 10:00 a.m. What fraction of the room had been painted by 9:30 a.m.? Express your answer as a common fraction

Connecting the centers of the four faces of a regular tetrahedron creates a smaller regular tetrahedron. What is the ratio of the volume of the smaller tetrahedron to the volume of the original one? Express your answer as a common fraction

If 45a = 54_b for positive integers a and b, what is the smallest possible value of a + b?

If point Q lies on side AB of square ABCD such that QC = \(\sqrt{10}\)units and QD = \(\sqrt{13}\) units, what is the area of square ABCD?

An equiangular hexagon has side lengths 3, 4, 5, 3, 4 and 5 units in that order. What is its area? Express your answer as a common fraction in simplest radical form. 

In the sum 4 + \(2\sqrt{2}\) + 2 + \(\sqrt{2}\) + … each term is obtained by dividing the previous term by \(\sqrt{2}\) . If the sum of the series, in simplest radical form, is m + \(n\sqrt{2}\) , what is the value of m + n?