➕➖ mathy ➗✖️
Welcome!✌ I write math stuff here.
☀️🌒
Latin StandardLibertinus
Pembahasan KSN Matematika SMA 2023 (Hari Pertama)
Published on 06 September 2023Contents
P1
Soal
An acute triangle $ABC$ has $BC$ as its longest side. Points $D,E$ respectively lie on $AC,AB$ such that $BA = BD$ and $CA = CE$. The point $A'$ is the reflection of $A$ against line $BC$. Prove that the circumcircles of $ABC$ and $A'DE$ have the same radii.Pembahasan
Hint
Buktikan kelima titik $A', B, C, D, E$ terletak pada satu lingkaran.
P2
Soal
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$: \[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]Pembahasan
Hint
Buktikan bahwa $\text{Im}(f)$ merupakan subset dari bilangan bulat.
Solusi
Denote $P(x,y)$ as the assertion of $x$ and $y$ to the given functional equation on the problem.
From $P(x, x - f(x))$, we obtain $f(x) = \lfloor \text{something} \rfloor$, so $\text{Im}(f) \subseteq \mathbb{Z}$.
From $P(f(x), 0)$ we obtain
\[ f(f(f(x))) = \lfloor f(x) + f(f(0)) \rfloor = f(x) + f(f(0)). \text{ }(\star) \]
And therefore, from comparing $P(x, f(y))$ and $P(y, f(x))$ we obtain:
\begin{align*}
\lfloor x + f(f(f(y))) \rfloor = f(f(x) + f(y)) = \lfloor y + f(f(f(x))) \rfloor &\stackrel{\text{Im}(f) \subseteq \mathbb{Z}}{\implies} \lfloor x \rfloor + f(f(f(y))) = \lfloor y \rfloor + f(f(f(x))) \\
&\stackrel{(\star)}{\implies} \lfloor x \rfloor + f(y) + f(f(0)) = \lfloor y \rfloor + f(x) + f(f(0)) \\
&\implies f(x) - \lfloor x \rfloor = K,
\end{align*}
for some constant $K$. Since $f(x)$ and $\lfloor x \rfloor$ are both integers, then $K$ must be an integer as well, so the solutions are of the form $\boxed{f(x)=\lfloor x \rfloor + K}$, where $K \in \mathbb{Z}$.
It should be easy to check this into the original equation.
P3
Soal
A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$, and replace $X$ with $X+d$ if Neneng chose the sign "up" or $X-d$ if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero.Prove that if $n \geq 14$, Asep can win in at most $(n-5)/4$ steps.
Pembahasan
Hint
tba
Solusi
tba
P4
Soal
Determine whether or not there exists a natural number $N$ which satisfies the following three criteria:
- $N$ is is divisible by $2^{2023}$, but not by $2^{2024}$,
- $N$ only has three different digits, and none of them are zero,
- Exactly $99.9%$ of the digits of $N$ are odd.
Pembahasan
Hint
tba
Solusi
tba
© 2021 donbasta - Huge thanks to vincentdoerig for the cool Latex style!