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 xx 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 x2x^{2}. Prove that the 2 first digits are 14......

Note by Samuraiwarm Tsunayoshi
5 years, 8 months ago

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I just figured no. 2 out. XD

Given that x2xxx2x(10n+1)x(10n+1)x^{2} | \overline{xx} \rightarrow x^{2} | x(10^{n}+1) \rightarrow x | (10^{n} + 1).

Factor 10n+110^{n} + 1 as a×ba\times b where a,ba, b are positive integers.

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

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

When we divide 10n+110^{n} + 1 by 7, it always starts with 1, 4 for n2n \geq 2. Hence, proven. ~~~

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

Samuraiwarm Tsunayoshi - 5 years, 8 months ago

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Let angle ACBACB's size be xx.

Then angle EDCEDC's and angle EDBEDB's size will be 90x90-x and xx, respectively.

From the properties, we will get that

BDEBCD\bigtriangleup BDE \sim \bigtriangleup BCD

Thus, BEBD=BDBC\frac{BE}{BD} = \frac{BD}{BC}

BE=BD2BC=AB2AD2AB+BDBE = \frac{BD^{2}}{BC} = \frac{AB^{2} - AD^{2}}{AB + BD} (Pythagorean Theorem)=ABAD=ACAD = AB - AD = AC - AD (Isosceles Triangle)=DC= DC Q.E.D.Q.E.D. ซ.ต.พ.

พิสูจน์สองข้อนี้ผมแป้กในห้องสอบทั้งคู่ครับ ไม่ทราบว่าได้เหรียญอะไรอ่อครับ

Sanchayapol Lewgasamsarn - 5 years, 6 months ago

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

Samuraiwarm Tsunayoshi - 5 years, 6 months ago

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ผมก็ได้แค่เหรียญทองแดงครับ เจอเรขานี้ Let it go แต่ข้อ Number Theory แสดงไปซุยๆครับแค่สองบรรทัดแรก 555

Sanchayapol Lewgasamsarn - 5 years, 5 months ago

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@Sanchayapol Lewgasamsarn เหมือนกันเลยครับ T__T แต่ข้อเรขา เรานั่งลุยถึกสมมติตัวแปร x แล้วไล่หาทุกด้าน แฮ่กๆๆๆ

Samuraiwarm Tsunayoshi - 5 years, 5 months ago

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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 - 5 years, 8 months ago

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