Found problems: 10
STEMS 2023 Math Cat A, 1
The following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$
Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board.
If the sum of all possible numbers she can end up with is $\dfrac{p}{q}$ where $p, q$ are coprime, then what
is the value of $\log_{2}(p+q)$?
STEMS 2023 Math Cat A, 4
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:
$\qquad\ (1) \; f(1) = 1.$
$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$
$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$
$\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$
Find the sum of all possible values of $f(2023)$.
STEMS 2023 Math Cat A, 2
Consider the set $S$ of permutations of $1, 2, \dots, 2022$ such that for all numbers $k$ in the
permutation, the number of numbers less than $k$ that follow $k$ is even.
For example, for $n=4; S = \{[3,4,1,2]; [3,1,2,4]; [1,2,3,4]; [1,4,2,3]\}$
If $|S| = (a!)^b$ where $a, b \in \mathbb{N}$, then find the product $ab$.
STEMS 2021 Math Cat B, Q5
Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid
$\begin{tabular}{|c|c|c|c|c|c|}
\hline
1 & 1 & 1 & 1 & 1 & 0\\
\hline
1 & 1 & 1 & 1 & 0 & 0\\
\hline
1 & 1 & 1 & 0 & 0 & 0\\
\hline
1 & 1 & 0 & 0 & 0 & 1\\
\hline
1 & 0 & 0 & 0 & 1 & 0\\
\hline
0 & 0 & 0 & 1 & 0 & 0\\
\hline
\end{tabular}$
and asked him to transform it to the new grid below
$\begin{tabular}{|c|c|c|c|c|c|}
\hline
1 & 2 & 18 &24 &28 &30\\
\hline
21 & 3 & 4 &16 &22 &26\\
\hline
23 &19 & 5 & 6 &14 &20\\
\hline
32 &25 &17 & 7 & 8 &12\\
\hline
33 &34 &27 &15 & 9 &10\\
\hline
35 &31 &36 &29 &13 &11\\
\hline
\end{tabular}$
by only applying the following algorithm:
$\bullet$ At each step, Sheldon must choose either two rows or two columns.
$\bullet$ For two columns $c_1, c_2$, if $a,b$ are entries in $c_1, c_2$ respectively, then we say that $a$ and $b$ are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.
Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.
STEMS 2023 Math Cat A, 8
For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?
STEMS 2023 Math Cat A, 7
Suppose a biased coin gives head with probability $\dfrac{2}{3}$. The coin is tossed repeatedly, if
it shows heads then player $A$ rolls a fair die, otherwise player $B$ rolls the same die. The process
ends when one of the players get a $6$, and that player is declared the winner.
If the probability that $A$ will win is given by $\dfrac{m}{n}$ where $m,n$ are coprime, then what is the value of $m^2n$?
STEMS 2023 Math Cat A, 5
A convex quadrilateral $ABCD$ is such that $\angle B = \angle D$ and are both acute angles. $E$ is
on $AB$ such that $CB = CE$ and $F$ is on $AD$ such that $CF = CD$. If the circumcenter of $CEF$ is
$O_1$ and the circumcenter of $ABD$ is $O_2$. Prove that $C,O_1,O_2$ are collinear.
[i]Proposed by Kapil Pause[/i]
STEMS 2023 Math Cat A, 3
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$, let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$, and $Q$ be the foot of the perpendicular from $B$ onto $AC$. Denote by $X$ the intersection point of the lines $FH$ and $QO$. Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$, then find the value of $1000a + 100b + 10c + d$.
STEMS 2023 Math Cat A, 5
Consider a polynomial $P(x) \in \mathbb{R}[x]$, with degree $2023$, such that $P(\sin^2(x))+P(\cos^2(x)) =1$ for all $x \in \mathbb{R}$. If the sum of all roots of $P$ is equal to $\dfrac{p}{q}$ with $p, q$ coprime, then what is the product $pq$?
STEMS 2023 Math Cat A, 6
There are $5$ vertices labelled $1,2,3,4,5$. For any two pairs of vertices $u, v$, the edge $uv$
is drawn with probability $1/2$. If the probability that the resulting graph is a tree is given by $\dfrac{p}{q}$ where $p, q$ are coprime, then find the value of $q^{1/10} + p$.