Difference between revisions of "2022 MMATHS Individual Round Problems"
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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>. | ||
| − | [[ | + | [[2022 MMATHS Individual Round Problems/Problem 1|Solution]] |
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| + | == Problem 2 == | ||
| + | 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== | ||
| + | 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== | ||
| + | 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 14:43, 19 December 2022
Problem 1
Suppose that 
and 
. Compute 
.
Problem 2
Triangle 
has 
and 
. Points 
and 
are the reflections of 
over 
, over , over 
, over , over , and over , respectively. Find the area of hexagon 
.
Problem 3
Luke and Carissa are finding the sum of the first 
positive integers by adding them one at a time. Luke forgets to add one number and gets an answer of 
. Carissa adds a number twice by mistake and gets an answer of 
. What is the sum of the number that Luke forgot and the number that Carissa added twice?
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?
Problem 5
- PLEASE NOTE THIS IS UNFINISHED.
