$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE$, $\angle CKD=\angle KFA.$
Prove that $KC=KF$
Search found 86 matches
- Sat May 07, 2016 8:25 pm
- Forum: Geometry
- Topic: China TST 2016 P1
- Replies: 3
- Views: 3441
- Wed May 04, 2016 7:54 pm
- Forum: Geometry
- Topic: A cool Geo!
- Replies: 1
- Views: 8856
A cool Geo!
In $\triangle ABC$, perpendicular bisector of sides $AB$ and $AC$ meet the internal bisector of $\angle BAC $ at $X$ and $Y$, respectively.
Prove that if circle $ACX$ touches $BC$ at $C$ and meets $AB$ again at $Z$ then $BZ=CA$ and circle $ABY$ touches $BC$ at $B$.
Prove that if circle $ACX$ touches $BC$ at $C$ and meets $AB$ again at $Z$ then $BZ=CA$ and circle $ABY$ touches $BC$ at $B$.
- Sat Apr 23, 2016 1:40 am
- Forum: Secondary Level
- Topic: 2014-national
- Replies: 15
- Views: 11817
Re: 2014-national
How? Can you please tell it?seemanta001 wrote:The solution is:
For an $m \times n$ chessboard, the number of the knights is $n \times \lceil\dfrac{m}{3}\rceil$.
- Wed Apr 20, 2016 1:25 pm
- Forum: Number Theory
- Topic: China TST 1987, problem 5
- Replies: 1
- Views: 2584
Re: China TST 1987, problem 5
Let us assume that $ x \geq y \geq z $ thus $ 3x^3 \geq x^3 + y^3 + z^3 = nx^{2}y^{2}z^{2} .$ So, $ x \geq \frac {ny^2z^2} {3} $,Because, $ x^{3} + y^{3} + z^{3} = nx^{2}y^{2}z^{2} $ thus,$ x^{2} | y^{3} + z^{3}$ , so $ 2y^{3} \geq y^{3} + z^{3} \geq x^{3} \geq \frac {n^2y^4z^4} {9} $ If $ z > 1 $ ...
- Mon Feb 15, 2016 7:03 pm
- Forum: Secondary Level
- Topic: Find p
- Replies: 1
- Views: 2670
Re: Find p
$p$ has the only solution 3.
Hint:Bertrand's postulate... Check this: https://en.wikipedia.org/wiki/Bertrand%27s_postulate
Hint:Bertrand's postulate... Check this: https://en.wikipedia.org/wiki/Bertrand%27s_postulate
- Sat Jan 09, 2016 9:06 pm
- Forum: Divisional Math Olympiad
- Topic: Dhaka Regional 2015 Higher Secondary/9 Secondary/9
- Replies: 1
- Views: 2725
Dhaka Regional 2015 Higher Secondary/9 Secondary/9
$ \Gamma _{1}$, $\Gamma _{2}$ are two circles with radius $\surd{2}$ and $2$ unit consecutively. The distance between their centres is $2\surd{2}$ unit. The two circle intersect at point $P$ and $Q$. $PR$ is a chord of $\Gamma _{2}$ which is bisected by $ \Gamma _{1}$. $PR$ =?
- Sat Jan 09, 2016 8:48 pm
- Forum: Junior: Solved
- Topic: Dhaka Junior 2015/9
- Replies: 1
- Views: 8155
Dhaka Junior 2015/9
$ABCD$ একটি সামান্তরিক যার কর্ণদ্বয়ের ছেদবিন্দু $O$। $AO ও BC$ এর মধ্যবিন্দু যথাক্রমে $P ও Q$. $\angle A =\angle DPQ$ এবং $\angle DBA =\angle DQP$। $AB$ এর দৈর্ঘ্য $1$ একক হলে, $ABCD$ এর ক্ষেত্রফল কত? $ABCD$ is a parallelogram and it's diagonals meet at point $O$. $P$ and $Q$ are the midpoints of $...
- Sat Jan 09, 2016 7:13 pm
- Forum: Junior: Solved
- Topic: Divisional MO-2015 (Kushtia) Junior P-10
- Replies: 2
- Views: 8203
Re: Divisional MO-2015 (Kushtia) Junior P-10
I think you didn't notice that the computer "Ramanujan", when multiplies $(4,7)$ then the result is $34.$ By the by, the answer is $36$. After multiplying, the computer converts the results in $8-base$ number. You can see that $3*5=15$. In, 8-base number, it'll be $17$.You can check for others multi...
- Wed Jan 06, 2016 10:12 pm
- Forum: Divisional Math Olympiad
- Topic: Kushtia-2015 Junior P-10
- Replies: 5
- Views: 4472
Re: Kushtia-2015 Junior P-10
Yes, I got that very early... But thank you
- Tue Jan 05, 2016 2:26 pm
- Forum: Number Theory
- Topic: National Number Theory
- Replies: 1
- Views: 2419
National Number Theory
সকল মৌলিক সংখ্যা $p$ এবং স্বাভাবিক সংখ্যা $a,b$ বের কর যেন $p^a + p^b $ একটি square number হয়।
Find all prime number $p$ and natural number $a,b$, such $p^a + p^b $ is a square number.
Find all prime number $p$ and natural number $a,b$, such $p^a + p^b $ is a square number.