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2023 SSMO Speed Round Problems

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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