2 Problems from math competition (OBEC IMSO) in Thailand.

I had a math comp. 3 days ago. And I can't do these problems. Stuck a lot. :<

1.) Triangle ABC given that AB = AC. Draw a line from B perpendicular to AC at point D. Draw another line from D perpendicular to BC at E. If BC = AB + AD, prove that BE = CD.

2.) Given a number $x$ with more than 1 digit. If we write it twice (such as x = 137, we write it 137137), we'll get a number that is divisible by $x^{2}$ . Prove that the 2 first digits are 14......

#Geometry #NumberTheory #MathCompetitions

Easy Math Editor

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`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I just figured no. 2 out. XD

Given that $x^{2} | \overline{xx} \rightarrow x^{2} | x(10^{n}+1) \rightarrow x | (10^{n} + 1)$ .

Factor $10^{n} + 1$ as $a\times b$ where $a, b$ are positive integers.

Since we know that both of them are odd numbers, but not divisible by 3 and 5 since $3, 5 \not| (10^{n} + 1)$ .

So $a, b$ must be divisible by 7.

When we divide $10^{n} + 1$ by 7, it always starts with 1, 4 for $n \geq 2$ . Hence, proven. ~~~

I always hate myself when I can't do it during the test but after the test. >:(

Samuraiwarm Tsunayoshi - 7 years, 3 months ago

Let angle $ACB$ 's size be $x$ .

Then angle $EDC$ 's and angle $EDB$ 's size will be $90-x$ and $x$ , respectively.

From the properties, we will get that

$\bigtriangleup BDE \sim \bigtriangleup BCD$

Thus, $\frac{BE}{BD} = \frac{BD}{BC}$

$BE = \frac{BD^{2}}{BC} = \frac{AB^{2} - AD^{2}}{AB + BD}$ (Pythagorean Theorem) $= AB - AD = AC - AD$ (Isosceles Triangle) $= DC$ $Q.E.D.$ ซ.ต.พ.

พิสูจน์สองข้อนี้ผมแป้กในห้องสอบทั้งคู่ครับ ไม่ทราบว่าได้เหรียญอะไรอ่อครับ

Sanchayapol Lewgasamsarn - 7 years, 1 month ago

ได้แค่เหรียญทองแดงเองคับ เสียไป 2 ข้อนี้ฟรีๆ เหมือนเส้นผมบังภูเขา T__T PS: ข้อแรกเพิ่งคิดได้ตอนนั่งคิดเล่นๆ ใช้เวลาแค่ 5 นาทีเสร็จ ==''

Samuraiwarm Tsunayoshi - 7 years, 1 month ago

ผมก็ได้แค่เหรียญทองแดงครับ เจอเรขานี้ Let it go แต่ข้อ Number Theory แสดงไปซุยๆครับแค่สองบรรทัดแรก 555

Sanchayapol Lewgasamsarn - 7 years ago

@Sanchayapol Lewgasamsarn – เหมือนกันเลยครับ T__T แต่ข้อเรขา เรานั่งลุยถึกสมมติตัวแปร x แล้วไล่หาทุกด้าน แฮ่กๆๆๆ

Samuraiwarm Tsunayoshi - 7 years ago

Hi Samuraiwarm!

Sorry, but I wasn't able to solve any of them. But could you please do me a favour?

Please send all the questions to me, I want to give all a try. My email address is sgsuper@yahoo.com, thanks.

Satvik Golechha - 7 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$