Problems From March 2019 Meeting


Problem 1. (Chosen by AP) Explain this paradox: the area of the rectangle is 104 square units, but if you add up the areas individual parts you get 20 + 52 + 24 + 7.5 = 103.5 square units:

Problem 2. (Chosen by AP) The sum of the positive divisors of 360 is 1170. What is the sum of the reciprocals of the positive divisors of 360?

Problem 3. (Chosen by AP) An hourglass is formed from two identical cones as shown. Initially, the upper cone is filled with sand and the lower one is empty. The sand flows at a constant rate from the upper cone to the lower cone. It takes exactly one hour to empty the upper cone. How long does it take for the depth of the sand in the lower cone to one half of the depth of sand in the upper cone? (Assume that the sand stays level in both cones at all times.)

Problem 4. (Chosen by VS) Two circles meet at P and Q. A line intersects segment PQ and meets the circles at the points A, B, C, D in that order. Prove that APB = ∠CQD.

Problem 5. (Chosen by VS) Solve the following equation:
(sinx + sin60°)/(sin(x + 60°)) = (2cosxsin(x + 30°))/(sin(2x + 30°))
Hint:
Product to Sum Sum to Product
2cosθcosφ = cos(θ − φ) + cos(θ + φ) sinθ±sinφ = 2sin(θ±φ)/(2)cos(θφ)/(2)
2sinθsinφ = cos(θ − φ) − cos(θ + φ) cosθ + cosφ = 2cos(θ + φ)/(2)cos(θ − φ)/(2)
2sinθcosφ = sin(θ + φ) + sin(θ − φ) cosθ − cosφ =  − 2sin(θ + φ)/(2)sin(θ − φ)/(2)
2cosθsinφ = sin(θ + φ) − sin(θ − φ)

Problem 6. (Chosen by MP) The triangle is equilateral. Show that for any line going through the vertex of the isosceles triangle, the sum of the radii of the two circles tangent to the sides of the triangle and the two lines, as shown in the diagram, is constant.

Problem 7. (Chosen by KK) Let M = 12345678910111213 … 20182019, be the string of numbers 1, 2, 3, …, 2018, 2019. Determine the remainder when M is divided by 45.

Problem 8. (Chosen by KK) A, B, C are three angles such that sinA + sinB + sinC = cosA + cosB + cosC = 0. Determine the value of cos2A + cos2B + cos2C.

Problem 9. (Chosen by MB) A rock is dropped into a water well and it travels approximately 16t2 in t seconds. If the splash is heard 3.5 seconds later and the speed of sound is 1087 feet/second, what is the depth of the well?

Problem 10. (Chosen by RA) Let ab = ab + a + b for all integers a and b. Evaluate 1⋆(2⋆(3⋆(4⋆⋯(99⋆100)))).

Problem 11. (Chosen by RA) Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of AB, and OA = 2(2). Segments OC and OD are tangent to the circles centered at A and B, respectively, and EF is a common tangent. What is the area of the shaded region ECODF?

Problem 12. (Chosen by MW) The points E(6,  4) and F(14,  12) lie in the standard (x,  y) coordinate plane shown below. Poind D lies on EF such that the length of EF is 4 times the length of DE. What are the coordinates of D?

Problem 13. (Chosen by AP) Choose 10 points on a circle so 3 chords with endpoints among these 10 points never meet at a common point inside the circle. Of the different triangles that can be created by chords connecting pairs of these 10 points, how many have no vertex on the circle?

Problem 14. (Chosen by MP & CPM)
Old Boniface he took his cheer,
Then he drilled a hole through a solid sphere -
Clear through the center, straight and strong.
And the whole was just six inches long.
Now tell us, when the end was gained
What volume in the sphere remained?
Sounds like we haven’t told enough,
But that’s all you need- and it isn’t tough.
 
OR: A cylindrical hole of length 1 meter is bored exactly through the center of a solid sphere of radius r. Show that the volume of the remaining hollow sphere is of a constant value and is independent of radius r.
Hint: The volume of a spherical cap of height k in a sphere of radius r is given by: V = (1)/(3)πk2(3r − k).

Problem 15. (Chosen by MP) The two pentagons are regular pentagons. Find the ratio of the two radii shown in the diagram.
Hint: cos30° = ((5) + 1)/(4), cos72° = ((5) − 1)/(4), cos108° = (1 − (5))/(4).

Problem 16. (Chosen by VS) Fermat numbers are numbers in the form of Fn = 22n + 1 where n is a nonnegative integer. They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 – 1665) who first studied numbers in this form. It is still an open problem to whether there are infinitely many primes in the form of Fn = 22n + 1 . The only know “Prime Fermat numbers” are F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537. Prove that:
  1. For n ≥ 1, Fn = (Fn − 1 –1)2 + 1.
  2. For n ≥ 1, Fn = F0Fn − 1 + 2.
  3. For n ≥ 2, the last digit of Fn is 7.

Problem 17. (Chosen by GB) Let ABCD be a square. Let M be a point on AB and let N be a point on BC such that MNMD. Show that
AMAB + CNCB = DM2.

Problem 18. (Chosen by MP) Let AB, BC be two adjacent sides of a regular nonagon inscribed in a circle with center in O. Let M be the midpoint of AB and N the midpoint of the radius perpendicular to BC. Show that the angle OMN = 30°.
Hint: The key is to find a cyclic quadrilateral.
Hint 2: If P is the intersection of ON with the circle, what can one say about the triangle AOP?

Problem 19. (Chosen by BW)
The sum of consecutive numbers are called the triangular numbers (each row increases by 1) (fig.1).
The sum of consecutive odd numbers are called the square numbers (fig.2).
The pentagonal numbers are the sequence
1,  5,  12,  22,  35,  …  (fig.3)
What is the difference between the 99th pentagonal number and the 100th?