February 24, 2020

State Competitions are coming up fast! Are you ready to compete? Let’s try a few 2019 State Competition problems to get ready.

2019 State Sprint Round, #18

If C is a digit such that the product of the three-digit numbers 2C8 and 3C1 is the five-digit number 90C58, what is the value of C?

Let’s work with just the rightmost two digits. For the units digit, 8 × 1 = 8 does not impact the tens digit. To get the tens digit of the product, we need to cross-multiply the units and tens digits of the two factors: C × 1 + 8 × C = 9C must end in 5. The only digit for C for which that works is C = 5

2019 State Target Round, #7

Andy has a cube of edge length 10 cm. He paints the outside of the cube red and then divides the cube into smaller cubes, each of edge length 1 cm. Andy randomly chooses one of the unit cubes and rolls it on a table. If the cube lands so that an unpainted face is on the bottom, touching the table, what is the probability that the entire cube is unpainted? Express your answer as a common fraction.

When a cube is subdivided along each of the three face-centered axes into n congruent slabs, a block composed of n3 congruent smaller cubes is formed. Each of those smaller cubes has 6 faces, resulting in a total of 6n3 faces. Only the outer surface – the 6 faces – of the original cube is painted. Each of the 6 faces of the original cube involves 1 face from each of the n2 smaller cubes making up the larger face, yielding 6n2 smaller faces that are painted, with the remaining 6n3 – 6n2 smaller faces unpainted. Removing the outer layer on each face yields an (n – 2) × (n – 2) × (n – 2) cube of totally unpainted smaller blocks, with 6(n – 2)3 unpainted smaller faces. Thus, with each of the 6n3 – 6n2 = 6(n3 – n2) unpainted smaller faces equally likely, of which 6(n – 2)3 correspond to completely unpainted smaller cubes, the probability of landing on a completely unpainted small cube upon landing on an unpainted smaller face is ((n - 2)3)/(n3 - n2). When n = 8, the probability is 63/(83 - 82) = 216/(512 - 64) = 216/448 = 27/56.

2019 State Team Round, #4

Suppose that Martians have eight fingers and use a base-eight (octal) number system. If Marty the Martian says he is 37 years old on Mars, how old is he in Earth’s base-ten system?

Just as 37 as a base-ten number means 3 × 101 + 7 × 100 = 3 × 10 + 7 × 1 = 37, so 37 as a base-eight number means 3 × 81 + 7 × 80 = 3 × 8 + 7 × 1 = 24 + 7 = 31 years in base ten. 

2019 State Countdown Round, #12

For a particular sequence, each term is the sum of the three preceding terms. If a, b, c, d, e, 0, 1, 2, 3 are consecutive terms of this sequence, what is the value of a + b + c + d + e?

As we’re told, each term is the sum of the three preceding terms. In order for this to be true, 2 = 1 + 0 + e, so e must equal 1. Similarly, 1 = 0 + e + d = 0 + 1 + d, so d = 0. Then, 0 = e + d + c = 1 + 0 + c, so c = -1. Continuing this pattern, we find that b = 2 and a = -1. Therefore, a + b + c + d + e = -1 + 2 + -1 + 0 + 1 = 1.

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