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| − | ==Problem 1==
| + | '''2023 SSMO Speed Round''' problems and solutions. The first link contains the full set of test problems. The second link contains the answer key. The rest contain each individual problem and its solution. |
| − | Let <math>S_1 = \{2,0,3\}</math> and <math>S_2 = \{2,20,202,2023\}.</math> Find the last digit of
| + | *[[2023 SSMO Speed Round Problems]] |
| − | \[
| + | *[[2023 SSMO Speed Round Answer Key]] |
| − | \sum_{a\in S_1,b\in S_2}a^b.
| + | **[[2023 SSMO Speed Round Problems/Problem 1|Problem 1]] |
| − | \]
| + | **[[2023 SSMO Speed Round Problems/Problem 2|Problem 2]] |
| − | [[2022 SSMO Speed Round Problems/Problem 1|Solution]] | + | **[[2023 SSMO Speed Round Problems/Problem 3|Problem 3]] |
| − | ==Problem 2==
| + | **[[2023 SSMO Speed Round Problems/Problem 4|Problem 4]] |
| − | Let <math>A</math>, <math>B</math>, <math>C</math> be independently chosen vertices lying in the square with coordinates <math>(-1, - 1)</math>, <math>(-1, 1)</math>, <math>(1, -1)</math>, and <math>(1, 1)</math>. The probability that the centroid of triangle <math>ABC</math> lies in the first quadrant is <math>\frac{m}{n}</math> for relatively prime positive integers <math>m</math> and <math>n.</math> Find <math>m+n.</math>
| + | **[[2023 SSMO Speed Round Problems/Problem 5|Problem 5]] |
| − | | + | **[[2023 SSMO Speed Round Problems/Problem 6|Problem 6]] |
| − | [[2022 SSMO Speed Round Problems/Problem 2|Solution]] | + | **[[2023 SSMO Speed Round Problems/Problem 7|Problem 7]] |
| − | ==Problem 3==
| + | **[[2023 SSMO Speed Round Problems/Problem 8|Problem 8]] |
| − | Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set <math>\{c,a,r,o,t\}.</math> Then, the pig randomly arranges the 6 letters to form a "word". If the 6 letters don't spell carrot, the pig gets frustrated and tries to spell it again (by rechoosing the 6 letters and respelling them). What is the expected number of tries it takes for the pig to spell "carrot"?
| + | **[[2023 SSMO Speed Round Problems/Problem 9|Problem 9]] |
| − | | + | **[[2023 SSMO Speed Round Problems/Problem 10|Problem 10]] |
| − | [[2022 SSMO Speed Round Problems/Problem 3|Solution]] | |
| − | ==Problem 4==
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| − | Let <math>F_1 = F_2 = 1</math> and <math>F_n = F_{n-1} + F_{n-2}</math> for all <math>n\geq 2</math> be the Fibonacci numbers. If distinct positive integers <math>a_1, a_2, \dots a_n</math> satisfies <math>F_{a_1}+F_{a_2}+\dots+F_{a_n}=2023</math>, find the minimum possible value of <math>a_1+a_2+\dots+a_n.</math>
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| − | [[2022 SSMO Speed Round Problems/Problem 4|Solution]] | |
| − | ==Problem 5==
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| − | In a parallelogram <math>ABCD</math> of dimensions <math>6\times 8,</math> a point <math>P</math> is choosen such that <math>\angle{APD}+\angle{BPC} = 180^{\circ}.</math> Find the sum of the maximum, <math>M</math>, and minimum values of <math>(PA)(PC)+(PB)(PD).</math> If you think there is no maximum, let <math>M=0.</math>
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| − | [[2022 SSMO Speed Round Problems/Problem 5|Solution]] | |
| − | ==Problem 6==
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| − | Find the smallest odd prime that does not divide <math>2^{75!} - 1</math>.
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| − | [[2022 SSMO Speed Round Problems/Problem 6|Solution]] | |
| − | ==Problem 7==
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| − | At FenZhu High School, <math>7</math>th graders have a 60\% of chance of having a dog and <math>8</math>th graders have a 40\% chance of having a dog. Suppose there is a classroom of <math>30</math> <math>7</math>th grader and <math>10</math> <math>8</math>th graders. If exactly one person owns a dog, then the probability that a <math>7</math>th grader owns the dog is <math>\frac{m}{n},</math> for relatively prime positive integers <math>m</math> and <math>n.</math> Find <math>m+n.</math>
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| − | [[2022 SSMO Speed Round Problems/Problem 7|Solution]] | |
| − | ==Problem 8==
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| − | Circle <math>\omega</math> has chord <math>AB</math> of length <math>18</math>. Point <math>X</math> lies on chord <math>AB</math> such that <math>AX = 4.</math> Circle <math>\omega_1</math> with radius <math>r_1</math> and <math>\omega_2</math> with radius <math>r_2</math> lie on two different sides of <math>AB.</math> Both <math>\omega_1</math> and <math>\omega_2</math> are tangent to <math>AB</math> at <math>X</math> and <math>\omega.</math> If the sum of the maximum and minimum values of <math>r_1r_2</math> is <math>\frac{m}{n},</math> find <math>m+n</math>.
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| − | [[2022 SSMO Speed Round Problems/Problem 8|Solution]] | |
| − | ==Problem 9==
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| − | Find the sum of the maximum and minimum values of <math>8x^2+7xy+5y^2</math> under the constraint that <math>3x^2+5xy+3y^2 = 88.</math>
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| − | [[2022 SSMO Speed Round Problems/Problem 9|Solution]] | |
| − | ==Problem 10==
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| − | In a circle centered at <math>O</math> with radius <math>7,</math> we have non-intersecting chords <math>AB</math> and <math>DC.</math> <math>O</math> is outisde of quadrilateral <math>ABCD</math> and <math>AB<CD.</math> Let <math>X = AO\cup CD</math> and <math>Y = BO\cup CD.</math> Suppose that <math>XO+YO = 7</math>. If <math>YC-DX=2</math> and <math>XY = 3</math>, then <math>AB = \frac{a\sqrt{b}}{c}</math> for <math>\gcd(a,c) = 1</math> and squareless <math>b.</math> Find <math>a+b+c.</math>
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| − | [[2022 SSMO Speed Round Problems/Problem 10|Solution]] | |
