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Pembahasan KSN Matematika SMA 2023 (Hari Kedua)
Published on 06 September 2021Contents
P5
Soal
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.Pembahasan
Hint
Tulis $a = dx$ dan $b = dy$, di mana $\text{gcd}(x, y) = 1$.
P6
Soal
Determine the number of permutations $a_1, a_2, \dots, a_n$ of $1, 2, \dots, n$ such that for every positive integer $k$ with $1 \le k \le n$, there exists an integer $r$ with $0 \le r \le n - k$ which satisfies \[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]Pembahasan
Hint
Perhatikan agar terdapat $k$ elemen yang jumlahnya $1 + 2 + \dots + k$,
$k$ buah elemen tersebut haruslah merupakan $1, 2, \dots, k$ by size argument.
Sehingga, $k+1$ memiliki dua kemungkinan posisi: di sebelah kiri atau kanan segmen
berisi $1, 2, \dots, k$ tersebut.
P7
Soal
Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.Pembahasan
Hint
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Solusi
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P8
Soal
Let $a, b, c$ be three distinct positive integers. Define $S(a, b, c)$ as the set of all rational roots of $px^2 + qx + r = 0$ for every permutation $(p, q, r)$ of $(a, b, c)$. For example, $S(1, 2, 3) = { -1, -2, -1/2 }$ because the equation $x^2+3x+2$ has roots $-1$ and $-2$, the equation $2x^2+3x+1=0$ has roots $-1$ and $-1/2$, and for all the other permutations of $(1, 2, 3)$, the quadratic equations formed don't have any rational roots.Determine the maximum number of elements in $S(a, b, c)$.
Pembahasan
Hint
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Solusi
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