MathDB

3

Part of 2014 ASDAN Math Tournament

Problems(8)

2014 Advanced #3

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7/1/2022
A mouse is playing a game of mouse hopscotch. In mouse hopscotch there is a straight line of 1111 squares, and starting on the first square the mouse must reach the last square by jumping forward 11, 22, or 33 squares at a time (so in particular the mouse’s first jump can be to the second, third, or fourth square). The mouse cannot jump past the last square. Compute the number of ways there are to complete mouse hopscotch.
2014Advanced Topics Test
2014 Advanced Tiebreaker #3

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7/1/2022
A robot is standing on the bottom left vertex (0,0)(0,0) of a 5×55\times5 grid, and wants to go to (5,5)(5,5), only moving to the right (a,b)↦(a+1,b)(a,b)\mapsto(a+1,b) or upward (a,b)↦(a,b+1)(a,b)\mapsto(a,b+1). However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path (a,b)↦(a−1,b+1)(a,b)\mapsto(a-1,b+1). As the grid is surrounded by walls, the robot cannot go outside the region 0≤a,b≤50\leq a,b\leq5. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
2014Advanced Topics Tiebreaker
2014 Algebra #3

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7/8/2022
Compute all prime numbers pp such that 8p+18p+1 is a perfect square.
2014Algebra Test
2014 Algebra Tiebreaker #3

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7/9/2022
Compute sin⁡(π9)sin⁡(2π9)sin⁡(4π9).\sin\left(\frac{\pi}{9}\right)\sin\left(\frac{2\pi}{9}\right)\sin\left(\frac{4\pi}{9}\right).
2014Algebra Tiebreaker
2014 General #3

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7/4/2022
Boris is driving on a remote highway. His car’s odometer reads 24942 km24942\text{ km}, which Boris notices is a palindromic number, meaning it is not changed when it is reversed. “Hm,” he thinks, “it should be a long time before I see that again.” But it takes only 11 hour for the odometer to once again show a palindromic number! How fast is Boris driving in km/h\text{km/h}?
2014General Test
2014 Team #3

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7/1/2022
A segment of length 11 is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region.
2014team test
2014 Geometry Tiebreaker #3

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7/1/2022
Let ABCABC be a triangle and II its incenter. Suppose AI=2AI=\sqrt{2}, BI=5BI=\sqrt{5}, CI=10CI=\sqrt{10} and the inradius is 11. Let A′A' be the reflection of II across BCBC, B′B' the reflection across ACAC, and C′C' the reflection across ABAB. Compute the area of triangle A′B′C′A'B'C'.
2014Geometry Tiebreaker
2014 Geometry #3

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7/12/2022
Compute the perimeter of the triangle that has area 3−33-\sqrt{3} and angles 45∘45^\circ, 60∘60^\circ, and 75∘75^\circ.
2014Geometry Test