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Pembahasan KSN Matematika SMA 2021 (Hari Kedua)

Contents

• P5
• P6
• P7
• P8

P5

Soal

Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.

Pembahasan

P6

Soal

There are $n$ natural numbers written on the board. Every move, we could erase $a,b$ and change it to $\gcd(a,b)$ and $\text{lcm}(a,b) - \gcd(a,b)$. Prove that in finite number of moves, all numbers in the board could be made to be equal.

Pembahasan

P7

Soal

Given $\triangle ABC$ with circumcircle $\ell$. Point $M$ in $\triangle ABC$ such that $AM$ is the angle bisector of $\angle BAC$. Circle with center $M$ and radius $MB$ intersects $\ell$ and $BC$ at $D$ and $E$ respectively, $(B \not= D, B \not= E)$. Let $P$ be the midpoint of arc $BC$ in $\ell$ that didn't have $A$. Prove that $AP$ angle bisector of $\angle DPE$ if and only if $\angle B = 90^{\circ}$.

Pembahasan

P8

Soal

On a $100 \times 100$ chessboard, the plan is to place several $1 \times 3$ boards and $3 \times 1$ board, so that

• Each tile of the initial chessboard is covered by at most one small board.
• The boards cover the entire chessboard tile, except for one tile.
• The sides of the board are placed parallel to the chessboard.

Suppose that to carry out the instructions above, it takes $H$ number of $1 \times 3$ boards and $V$ number of $3 \times 1$ boards. Determine all possible pairs of $(H,V)$.

Pembahasan

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