2022 MMATHS Individual Round Problems

Problem 1

Suppose that $a+b = 20, b+c = 22,$ and $a+c = 2022$ . Compute $\frac {a-b}{c-a}$ .

Problem 2

Triangle $ABC$ has $AB = 3, BC = 4,$ and $CA = 5$ . Points $D, E, F, G, H,$ and $I$ are the reflections of $A$ over $B$ , $B$ over $A$ , $B$ over $C$ , $C$ over $B$ , $C$ over $A$ , and $A$ over $C$ , respectively. Find the area of hexagon $EFIDGH$ .

Solution

Problem 3

Luke and Carissa are finding the sum of the first $20$ positive integers by adding them one at a time. Luke forgets to add one number and gets an answer of $207$ . Carissa adds a number twice by mistake and gets an answer of $225$ . What is the sum of the number that Luke forgot and the number that Carissa added twice?

Solution

Problem 4

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit perfect square!"

Claire asks, "If I picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there a chance I'd know for certain what it is?"

Cat says, "Yes!" Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn't know my favorite number!

Claire says, "Now I know your favorite number!" What is Cat's favorite number?

PLEASE NOTE THIS IS UNFINISHED.

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2022 MMATHS Individual Round Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4