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Difference between revisions of "2022 MMATHS Individual Round Problems"

(Created page with "2022 MMATHS Individual Round 2022 MMATHS Individual Round Problems Problem 1 Suppose that <math>a+b = 20, b+c = 22,</math> and <math>a+c = 2022</math>. Compute <math>\frac {a-...")
 
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2022 MMATHS Individual Round
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== Problem 1 ==
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2022 MMATHS Individual Round Problems
 
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Problem 1
 
 
Suppose that <math>a+b = 20, b+c = 22,</math> and <math>a+c = 2022</math>. Compute <math>\frac {a-b}{c-a}</math>.
 
Suppose that <math>a+b = 20, b+c = 22,</math> and <math>a+c = 2022</math>. Compute <math>\frac {a-b}{c-a}</math>.
  
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Solution
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[[2022 MMATHS Individual Round Problems/Problem 1|Solution]]
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== Problem 2 ==
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Triangle <math>ABC</math> has <math>AB = 3, BC = 4,</math> and <math>CA = 5</math>. Points <math>D, E, F, G, H, </math> and <math>I</math> are the reflections of <math>A</math> over <math>B</math>, <math>B</math> over <math>A</math>, <math>B</math> over <math>C</math>, <math>C</math> over <math>B</math>, <math>C</math> over <math>A</math>, and <math>A</math> over <math>C</math>, respectively. Find the area of hexagon <math>EFIDGH</math>. 
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[[2022 MMATHS Individual Round Problems/Problem 2|Solution]]
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==Problem 3==
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Luke and Carissa are finding the sum of the first <math>20</math> positive integers by adding them one at a time. Luke forgets to add one number and gets an answer of <math>207</math>. Carissa adds a number twice by mistake and gets an answer of <math>225</math>. What is the sum of the number that Luke forgot and the number that Carissa added twice?
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[[2022 MMATHS Individual Round Problems/Problem 3|Solution]]
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==Problem 4==
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Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit perfect square!"
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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?"
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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!
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Claire says, "Now I know your favorite number!" What is Cat's favorite number?
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[[2022 MMATHS Individual Round Problems/Problem 4|Solution]]
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==Problem 5==
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*PLEASE NOTE THIS IS UNFINISHED.

Latest revision as of 13:43, 19 December 2022

Problem 1

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

Solution

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?

Solution

Problem 5

  • PLEASE NOTE THIS IS UNFINISHED.
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