## 2022 VerityMC Solutions

Below are sample problems from the first VerityMC math competition held in 2021.

Given the high difficulty of the 2021 VerityMC, we adjusted difficulty of the 2022 competition to make the problems more enjoyable for participants.

## Speed Round #4

Charles has \$310 in his bank account, while Rohit has \$170 in his bank account. One day, they both decide to give 10% of their money to charity. How much more money than Rohit does Charles have after their donations?

## Speed Round #15

Justin is creating a mixture of coffee and cream to drink before a math competition. He decides to brew 8 ounces of coffee. He then pours 50% of the coffee into a cup containing 4 ounces of cream and mixes it thoroughly. Then, he pours 30% of that mixture into the original cup. What percentage of this final mixture is cream? Round to the nearest whole percent.

## Speed Round #4 Solution

If Charles gives 10% of his money to charity, he is giving away 310·10% = \$31. That means Charles has 310 − 31 = 279 dollars left.

If Rohit gives away 10% of his money, he loses \$170 · 10% = \$17. That means that Rohit has \$170−\$17 = \$153.

Then, subtract Rohit’s new balance from Charles’s balance. \$279 − \$153 = \$126

## Speed Round #15 Solution

Solution: We start with 8 ounces of coffee. 50% of 8 is 4. This means that Justin pours 4 ounces of coffee and mixes it with the 4 ounces of cream, creating a mixture that is 50% cream (4 oz coffee, 4 oz cream).

Then, when he pours 30% of the mixture into the cup, he is pouring 0.3 × 8 = 2.4 oz of cream and 0.3 × 8 = 2.4 oz of coffee into the original cup, creating a 8.8 ounce mixture of coffee and cream.

Since 2.4 of the 8.8 ounces of the mixture is cream, we divide 2.4 by 8.8 to get 0.2727. . . .

Converting this decimal to a percentage and rounding it to the nearest percent, we find out that approximately 27% of the final mixture is cream.

## Accuracy Round #2

I am thinking of a two-digit number whose tens digit is larger than the ones digit. The tens digit is even, while the ones digit is odd. If I subtract 10 from my number, the new number will be less than 20. What is my original number?

## Accuracy Round #5

Alex has an infinite number of coconuts numbered from 1, 2, 3, · · · . He colors them either green or red according to a color scheme. If the coconut’s number is a multiple of 3, 5, or 7, it is colored red. Otherwise, it is colored green. Find the number of the 100th coconut colored green.

## Accuracy Round #2 Solution

Let our number be x. We know that 10 subtracted from x is less than 20, which means x < 30. The tens digit of x must be even, and we are left with only 2 as a possibility.

The only two numbers with tens digit 2 and a smaller ones digit are 20 and 21. 21 is the only one with an odd ones digit.

## Accuracy Round #5 Solution

The LCM of 3, 5, and 7 is 105. We can find that the number of green coconuts of the first 105 is 23·45·67· 105 = 48 green coconuts.

The cycle repeats itself every 105 numbers, so there are 96 green coconuts in the first 105 · 2 = 210 coconuts.

Counting the next 4 coconuts, we find the numbers 211, 212, 214, and 218. 218 is the 100th green coconut.