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Difference between revisions of "2023 SSMO Speed Round Problems"
 
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==Problem 1==
 
==Problem 1==
 +
 
Let <math>S_1 = \{2,0,3\}</math> and <math>S_2 = \{2,20,202,2023\}.</math> Find the last digit of
 
Let <math>S_1 = \{2,0,3\}</math> and <math>S_2 = \{2,20,202,2023\}.</math> Find the last digit of
 
<cmath>\sum_{a\in S_1,b\in S_2}a^b.</cmath>
 
<cmath>\sum_{a\in S_1,b\in S_2}a^b.</cmath>
 +
 
[[2023 SSMO Speed Round Problems/Problem 1|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 1|Solution]]
 +
 
==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>
 
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 2|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 2|Solution]]
 +
 
==Problem 3==
 
==Problem 3==
 +
 
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"?
 
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 3|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 3|Solution]]
 +
 
==Problem 4==
 
==Problem 4==
 +
 
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>
 
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>
  
 
[[2023 SSMO Speed Round Problems/Problem 4|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 4|Solution]]
 +
 
==Problem 5==
 
==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>
 
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>
  
 
[[2023 SSMO Speed Round Problems/Problem 5|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 5|Solution]]
 +
 
==Problem 6==
 
==Problem 6==
 +
 
Find the smallest odd prime that does not divide <math>2^{75!} - 1</math>.
 
Find the smallest odd prime that does not divide <math>2^{75!} - 1</math>.
  
 
[[2023 SSMO Speed Round Problems/Problem 6|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 6|Solution]]
 +
 
==Problem 7==
 
==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>
 
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>
  
 
[[2023 SSMO Speed Round Problems/Problem 7|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 7|Solution]]
 +
 
==Problem 8==
 
==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>.
 
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>.
  
 
[[2023 SSMO Speed Round Problems/Problem 8|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 8|Solution]]
 +
 
==Problem 9==
 
==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>  
 
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>  
  
 
[[2023 SSMO Speed Round Problems/Problem 9|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 9|Solution]]
 +
 
==Problem 10==
 
==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>
 
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>
  
 
[[2023 SSMO Speed Round Problems/Problem 10|Solution]]
 
[[2023 SSMO Speed Round Problems/Problem 10|Solution]]

Latest revision as of 15:24, 2 May 2025

Problem 1

Let $S_1 = \{2,0,3\}$ and $S_2 = \{2,20,202,2023\}.$ Find the last digit of \[\sum_{a\in S_1,b\in S_2}a^b.\]

Solution

Problem 2

Let $A$, $B$, $C$ be independently chosen vertices lying in the square with coordinates $(-1, - 1)$, $(-1, 1)$, $(1, -1)$, and $(1, 1)$. The probability that the centroid of triangle $ABC$ lies in the first quadrant is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 3

Pigs like to eat carrots. Suppose a pig randomly chooses 6 letters from the set $\{c,a,r,o,t\}.$ 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"?

Solution

Problem 4

Let $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n\geq 2$ be the Fibonacci numbers. If distinct positive integers $a_1, a_2, \dots a_n$ satisfies $F_{a_1}+F_{a_2}+\dots+F_{a_n}=2023$, find the minimum possible value of $a_1+a_2+\dots+a_n.$

Solution

Problem 5

In a parallelogram $ABCD$ of dimensions $6\times 8,$ a point $P$ is choosen such that $\angle{APD}+\angle{BPC} = 180^{\circ}.$ Find the sum of the maximum, $M$, and minimum values of $(PA)(PC)+(PB)(PD).$ If you think there is no maximum, let $M=0.$

Solution

Problem 6

Find the smallest odd prime that does not divide $2^{75!} - 1$.

Solution

Problem 7

At FenZhu High School, $7$th graders have a 60\% of chance of having a dog and $8$th graders have a 40\% chance of having a dog. Suppose there is a classroom of $30$ $7$th grader and $10$ $8$th graders. If exactly one person owns a dog, then the probability that a $7$th grader owns the dog is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 8

Circle $\omega$ has chord $AB$ of length $18$. Point $X$ lies on chord $AB$ such that $AX = 4.$ Circle $\omega_1$ with radius $r_1$ and $\omega_2$ with radius $r_2$ lie on two different sides of $AB.$ Both $\omega_1$ and $\omega_2$ are tangent to $AB$ at $X$ and $\omega.$ If the sum of the maximum and minimum values of $r_1r_2$ is $\frac{m}{n},$ find $m+n$.

Solution

Problem 9

Find the sum of the maximum and minimum values of $8x^2+7xy+5y^2$ under the constraint that $3x^2+5xy+3y^2 = 88.$

Solution

Problem 10

In a circle centered at $O$ with radius $7,$ we have non-intersecting chords $AB$ and $DC.$ $O$ is outisde of quadrilateral $ABCD$ and $AB<CD.$ Let $X = AO\cup CD$ and $Y = BO\cup CD.$ Suppose that $XO+YO = 7$. If $YC-DX=2$ and $XY = 3$, then $AB = \frac{a\sqrt{b}}{c}$ for $\gcd(a,c) = 1$ and squareless $b.$ Find $a+b+c.$

Solution

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