ข้อสอบ SMO 2017 Grade 10 (Eng)

[Pitchayut]

เดี๋ยวเฉลยจะทยอยพิมพ์ให้ทีหลังครับ ตอนนี้ทำได้เกือบทุกข้อแล้ว

Day 1

Problem 1. Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$. If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$, then define $f$ as an "$n$-variable Boolean function" and denote
$$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}.$$
(i) Determine the number of $n$-variable Boolean functions;
(ii) Let $g$ be a $10$-variable Boolean function satisfying
$$g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}$$ Evaluate the size of the set $D_{10} (g)$ and the following sum. $$\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$$

Problem 2. Let $ABC$ be an acute-angled triangle. In $ABC$, $AB \neq AB$, $K$ is the midpoint of the the median $AD$, $DE \perp AB$ at $E$, $DF \perp AC$ at $F$. The lines $KE$, $KF$ intersect the line $BC$ at $M$, $N$, respectively. The circumcenters of $\triangle DEM$, $\triangle DFN$ are $O_1, O_2$, respectively.

Prove that $O_1 O_2 \parallel BC$.

Problem 3. For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 1, 2$.
Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.

Problem 4. Real numbers $a_1, a_2, \cdots, a_{2017}$ satisfy $a_1 = a_{2017}$, and
$$|a_i + a_{i+2} - 2a_{i + 1}| \leq 1\qquad, i = 1, 2, \cdots, 2015.$$ Determine the maximum possible value of $\max\limits_{1 \leq i < j \leq 2017} |a_i - a_j|$.

Day 2

Problem 5. Let $ABCD$ be a cyclic quadrilateral with the circumcenter $O$. In $ABCD$, the diagonals $AC$, $BD$ are perpendicular to each other, $M$, $N$ are the midpoints of the arcs $\widehat{ADC}$, $\widehat{ABC}$, respectively, the diameter of its circumcircle through $D$ intersect the chord $AN$ at $G$. $K$ is a point on the segment $CD$ satisfying $GK \parallel NC$.

Prove that $BM \perp AK$.

Problem 6. The sequence $\{a_n\}$ satisfies $a_1 = \frac{1}{2}$, $a_2 = \frac{3}{8}$, and $a_{n + 1}^2 + 3 a_n a_{n + 2} = 2 a_{n + 1} (a_n + a_{n + 2}) \quad(n \in \mathbb{N^*})$.
$(1)$ Determine the general formula of the sequence $\{a_n\}$;
$(2)$ Prove that for any positive integer $n$, there is $0 < a_n < \frac{1}{\sqrt{2n + 1}}$.

Problem 7. Let $m$ be a positive integer, for $k = 1, 2, \cdots$, define $a_k = \dfrac{(2km)!}{3^{(k - 1)m}}$.
Prove that in the sequence $a_1, a_2, \cdots$, there are both infinitely many integer terms and non-integer terms.

Problem 8. Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.

[nowhere]

รบกวนขอ HINT หน่อยครับ

[rendv]

Problem 5. (ทำซ่อนไม่เป็นอ่ะครับ 55)
ให้ $\angle BAC= \alpha$ และ $\angle NAB=\beta $
จาก $\angle ADC = 180^{\circ}-\angle ANC = 180^{\circ}-\angle AGK$ ดังนั้น $AGKD$ cyclic
จาก $N$ และ $M$ เป็นจุดกึ่งกลางส่วนโค้ง $ABC$ และ $ADC$ ตามลำดับ จะได้ว่า $\overline{NM}\bot \overline{AC}$ และได้ว่า $\overline{NM} \parallel \overline{BD}$
ทำให้ ส่วนโค้ง $HN =$ ส่วนโค้ง $MD =$ ส่วนโค้ง $NB$ และทำให้ส่วนโค้ง $AH =$ ส่วนโค้ง $BC$ ตามมา
จากข้อมูลส่วนนี้ เราสามารถไล่มุม $\alpha$ และ $\beta$ ได้ดังรูป

$\angle ACD = 90^{\circ}-\alpha$
$\angle CKG = 180^{\circ}-\angle NCK=180^{\circ}-\angle NAD=90^{\circ}-\beta\rightarrow \angle AKC=90^{\circ}-\beta+\alpha$
ดังนั้น $\angle CAK=180^{\circ}-(90^{\circ}-\alpha)-(90^{\circ}-\beta+\alpha)=\beta$
ทำให้ $APQM$ cyclic และได้ว่า $\angle AQM = \angle APM = 90^{\circ}$