MathDB

3

Part of 2015 ASDAN Math Tournament

Problems(8)

2015 Advanced #3

Source:

7/8/2022
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.
2015Advanced Topics Test
2015 Advanced Tiebreaker #3

Source:

7/8/2022
You have a circular necklace with 1010 beads on it, all of which are initially unpainted. You randomly select 55 of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?
2015Advanced Topics Tiebreaker
2015 Algebra #3

Source:

7/1/2022
Let a1,a2,a3,…,a6a_1,a_2,a_3,\dots,a_6 be an arithmetic sequence with common difference 33. Suppose that a1a_1, a3a_3, and a6a_6 also form a geometric sequence. Compute a1a_1.
2015Algebra Test
2015 Algebra Tiebreaker #3

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7/1/2022
Let f(x)f(x) be a polynomial of finite degree satisfying (x+9)f(x+1)=(x+3)f(x+3)(x+9)f(x+1)=(x+3)f(x+3) for all real xx. If f(0)=1f(0)=1, find the value of f(1)f(1).
2015Algebra Tiebreaker
2015 Guts #3

Source:

7/28/2022
Consider a unit circle with center OO. Let PP be a point outside the circle such that the two line segments passing through PP and tangent to the circle form an angle of 60∘60^\circ. Compute the length of OPOP.
2015Guts Test
2015 Geometry #3

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7/1/2022
Points EE and FF are chosen on sides BCBC and CDCD respectively of rhombus ABCDABCD such that AB=AE=AF=EFAB=AE=AF=EF, and FC,DF,BE,EC>0FC,DF,BE,EC>0. Compute the measure of ∠ABC\angle ABC.
2015Geometry Test
2015 Geometry Tiebreaker #3

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7/20/2022
Place points AA, BB, CC, DD, EE, and FF evenly spaced on a unit circle. Compute the area of the shaded 1212-sided region, where the region is bounded by line segments ADAD, DFDF, FBFB, BEBE, ECEC, and CACA.
2015Geometry Tiebreaker
2015 Team #3

Source:

8/3/2022
Simplify 7+33−7−33\sqrt{7+\sqrt{33}}-\sqrt{7-\sqrt{33}}.
2015team test