2024 SSMO Team Round Problems

Problem 1

How many ordered triples of positive integers $(a, b, c)$ satisfy the equation $2(a^b)^c+1=513$ ?

Problem 2

Find the sum of the three smallest positive integers $n$ where the last two digits of $n^4$ are $01$ .

Problem 3

Consider positive integers $N$ such that when $N$'s units digit and leading nonzero digit are removed, what remains is a two-digit perfect square. The average of all $N$ can be expressed as $\frac{m}{n},$ for relatively prime positive integers $m$ and $n.$ Find $m+n.$

Solution

Problem 4

Let $ABC$ be a right triangle with circumcenter $O$ and incenter $I$ such that $\angle ABC = 90^{\circ}$ and $\frac{AB}{BC} = \frac{3}{4}$ . Let $D$ the projection of $O$ onto $AB$ , and let $E$ be the projection of $O$ onto $BC$ . Denote $\omega_{1}$ be the incenter of $ADO$ and $\omega_{2}$ as the incenter of $OEC$ . If $\frac{[\omega_{1}\omega_{2}I]}{[ABC]}=\frac{m}{n},$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 5

Let $ABC$ be a triangle with $AB=AC=5$ and $BC=6$ . Let $\omega_1$ be the circumcircle of $ABC$ and let $\omega_2$ be the circle externally tangent to $\omega_1$ and tangent to rays $AB$ and $AC$ . If the distance between the centers of $\omega_1$ and $\omega_2$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m+n$ .

Solution

Problem 6

Let $\alpha$ , $\beta$ , and $\gamma$ be the roots of the polynomial $x^3 - 6x^2 - 19x - n$ . If $n$ is an integer, what is the least possible positive value of $\alpha^3 + \beta^3 + \gamma^3$ ?

Solution

Problem 7

Let $a$ and $b$ be real numbers that satisfy $a^3+8ab^2=8b^3+4a^2b=375$ . Find $\lfloor ab \rfloor$ .

Solution

Problem 8

Three integers $0\le a\le b\le c< 229$ satisfy the congruence $n^3 \equiv 1 \pmod{229}.$ Given that $71^2 - 3$ and $107^2 + 1$ are both multiples of $229,$ find the value of $b.$

Solution

Problem 9

Let $ABCDEFGH$ be an equiangular octagon such that $AB=6, BC=8, CD=10, DE=12, EF=6, FG=8, GH=10,$ and $AH=12$ . The radius of the largest circle that fits inside the octagon can be expressed as $a+b\sqrt{c},$ where $a,b,$ and $c$ are integers and $c$ is squarefree. Find $a+b+c.$

Solution

Problem 10

The side-lengths of a convex cyclic quadrilateral $ABCD$ are integers and $(AB\cdot AD+BC\cdot CD)^2=AC^2\cdot BD^2-72$ . Find all possible values of the perimeter of $ABCD$ .

Solution

Problem 11

Let $S$ denote the set of positive divisors of $5400.$ Let $S_i = \{d \mid d \in S, \, d \equiv i \pmod4\}$ and let $s_i$ denote the sum of all elements of $S_i.$ Find the value of $s_0^2+s_1^2+s_2^2+s_3^2-2s_0s_2-2s_1s_3.$

Solution

Problem 12

What is the smallest positive integer $n$ with 3 positive prime factors such that for all integers $k$ , $k^n \equiv k \pmod n$ ?

Solution

Problem 13

In a deck of 54 cards (2 identical jokers, 4 identical cards with $1,2,3,\dots,13$ ), each card is dealt to one of 3 people, each having a $\frac{1}{3}$ chance of receiving each card. If the expected sum of the number of unique cards the three of them have can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n,$ find $m+n.$

Solution

Problem 14

Let $a_1, a_2, \dots, a_7$ be the roots of the polynomial $x^7+5x^6+9x^5+x^4+x^3+10x^2+5x+1.$ Find the value of $\left|\prod_{n=1}^7 \prod_{m=n+1}^7 (a_na_m-1)\right|.$ Solution

Problem 15

In triangle $ABC$ inscribed in circle $\omega,$ let $M$ be the midpoint of $BC.$ Denote $P$ as the intersection of $AM$ with $\omega.$ If $BP = 9, CP = 13,$ and $AM = 20,$ find the perimeter of triangle $ABC.$