HEXA Challenge

Problems created by: Dr. Oscar Tagiyev

September Challenge:

A doctor prescribed a patient Tablet A to be taken 3 times a month. Another doctor prescribed the same patient Tablet B, to be taken 5 times every 2 months. The patient continues to take both tablets for a year. How many times will the patient have to take the tablets at the same time during the year?

October Challenge:

Inside a square, with each side equal to 1 unit, there are two circles with corresponding radii a and b.

The sum of diameters of the circles is less than 1 unit. What is the probability that the circles do not share a common point?

November Challenge:

A father has \$98,000 in his bank account. He decides to give his daughter and son some of the money. He asks each of his children how much money they would like. His daughter replies: Pay my brother first, I would be happy with 20% of whatever is left over.  His son replies: I’m not as greedy, pay my sister first, I would be happy with 10% of whatever remains.  Given his children’s answers, how much will the father have after paying each of his children?

December Challenge:

A circle inside of a triangle that is tangent to all three sides is an inscribed circle. Let’s define an outscribed circle as a circle outside of a triangle that is tangent to only one of its sides but is also tangent to the continuation of the other two sides. The triangle has three outscribed circles. Given their radii Ra , Rb,  Rc, what will be the triangle’s area?

January Challenge:

The base of a cylinder is a circle with radius, R. The circle has its
center at the origin point of the XY coordinate plane. The cylinder
has the height of H. If the plane containing the points (-R; 0,0),
(R; 0; H), (0; R; H/2), intersects the cylinder, what will be the
area of the intersection?

February Challenge:

How many pairs of integer
numbers (N; M)
satisfy the equation?

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