|
|
| Line 1: |
Line 1: |
| − | ==Problem 1==
| + | '''2023 SSMO Speed Round''' problems and solutions. '''This test was held on ZOOM on 6/27/2021.''' The first link contains the full set of test problems. 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]] |
| − | <cmath>\sum_{a\in S_1,b\in S_2}a^b.</cmath>
| + | *[[2023 SSMO Speed Round Answer Key]] |
| − | [[2022 SSMO Speed Round Problems/Problem 1|Solution]] | + | **[[2023 SSMO Speed Round Problems/Problem 1|Problem 1]] |
| − | ==Problem 2==
| + | **[[2023 SSMO Speed Round Problems/Problem 2|Problem 2]] |
| − | 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 3|Problem 3]] |
| − | | + | **[[2023 SSMO Speed Round Problems/Problem 4|Problem 4]] |
| − | [[2022 SSMO Speed Round Problems/Problem 2|Solution]] | + | **[[2023 SSMO Speed Round Problems/Problem 5|Problem 5]] |
| − | ==Problem 3==
| + | **[[2023 SSMO Speed Round Problems/Problem 6|Problem 6]] |
| − | 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 7|Problem 7]] |
| − | | + | **[[2023 SSMO Speed Round Problems/Problem 8|Problem 8]] |
| − | [[2022 SSMO Speed Round Problems/Problem 3|Solution]] | + | **[[2023 SSMO Speed Round Problems/Problem 9|Problem 9]] |
| − | ==Problem 4==
| + | **[[2023 SSMO Speed Round Problems/Problem 10|Problem 10]] |
| − | 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>
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 4|Solution]] | |
| − | ==Problem 5==
| |
| − | 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>
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 5|Solution]] | |
| − | ==Problem 6==
| |
| − | Find the smallest odd prime that does not divide <math>2^{75!} - 1</math>.
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 6|Solution]] | |
| − | ==Problem 7==
| |
| − | 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>
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 7|Solution]] | |
| − | ==Problem 8==
| |
| − | 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>.
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 8|Solution]] | |
| − | ==Problem 9==
| |
| − | 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>
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 9|Solution]] | |
| − | ==Problem 10==
| |
| − | 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>
| |
| − | | |
| − | [[2022 SSMO Speed Round Problems/Problem 10|Solution]] | |
