Search found 110 matches
- Thu Dec 10, 2015 8:20 pm
- Forum: Geometry
- Topic: 2013 All-Russian Olympiad Final Round Grade 10 Day 2 P7
- Replies: 3
- Views: 3567
Re: 2013 All-Russian Olympiad Final Round Grade 10 Day 2 P7
I haven't really tried out this problem, but it seems that $AA_2,BB_2,CC_2$ are concurrent. Furthermore, it seems(this one I am not so sure about) that $\Delta ABC$ is similar to $\Delta A_2B_2C_2$ , but not directly.
- Wed Dec 09, 2015 2:44 pm
- Forum: Divisional Math Olympiad
- Topic: Dhaka 2014,Secondary,P6
- Replies: 3
- Views: 2997
Re: Dhaka 2014,Secondary,P6
Yes, edited, thanks.tanmoy wrote:I think you wanted to tell that $D$ is the centre of the circle with centre on $BC$.rah4927 wrote:Let $D$ be the centre of the circle with centre on $AB$.
- Tue Dec 08, 2015 8:24 pm
- Forum: Divisional Math Olympiad
- Topic: Dhaka 2014,Secondary,P6
- Replies: 3
- Views: 2997
Re: Dhaka 2014,Secondary,P6
Since we obviously have to work with areas and perimeters, we might as well try and figure out the area and perimeter of the triangle. Let $D$ be the centre of the circle with centre on $CB$. Note that by joining A with D, we get the area of the triangle $\Delta ABC$ to be $K=9(b+c)$. Continue this ...
- Tue Dec 08, 2015 8:18 pm
- Forum: Number Theory
- Topic: 2001 Balkan Mathematical Olympiad
- Replies: 2
- Views: 2836
- Tue Dec 08, 2015 4:03 pm
- Forum: Number Theory
- Topic: Representing $m$ numbers using partial sums of at most $2^m$
- Replies: 1
- Views: 2105
Re: Representing $m$ numbers using partial sums of at most $
Some hints. This is a problem where you have to represent some numbers as a sum of some other numbers. We also need any partial sum to be different. This means that we need the $b_i$ to be a sequence of numbers so that adding any subsequence gives a distinct sum, and not only that, we need to be abl...
Re: পাকা চুল
See here
- Sat Dec 05, 2015 1:19 pm
- Forum: Number Theory
- Topic: Representing $m$ numbers using partial sums of at most $2^m$
- Replies: 1
- Views: 2105
Representing $m$ numbers using partial sums of at most $2^m$
(1220 Number Theory Problems) Let $m$ positive integers $a_1, \cdots, a_m$ be given. Prove that there exist fewer than $2^m$ positive integers $b_1, \cdots, b_n$ such that all sums of distinct $b_k$s are distinct and all $a_i (i \le m)$ occur among them.
- Fri Dec 04, 2015 6:13 pm
- Forum: Geometry
- Topic: Balkan MO 2014,G5
- Replies: 1
- Views: 2304
Re: Balkan MO 2014,G5
Prove that $E$ is the incentre of $\Delta DKL$ .
- Fri Nov 27, 2015 4:26 pm
- Forum: Combinatorics
- Topic: USAMO 2003 Problem 2
- Replies: 2
- Views: 2756
Re: USAMO 2003 Problem 2
Some hints. When dealing with rationals and irrationals in combinatorial geometry, law of cosines and law of sines are powerful tools(so is coordinate geometry, especially complex numbers and bary, but they are for another day). Can you prove the result for a convex quadrilateral? Can you now extend...
- Fri Nov 27, 2015 4:22 pm
- Forum: Combinatorics
- Topic: USAMO 2003 Problem 2
- Replies: 2
- Views: 2756
USAMO 2003 Problem 2
A convex polygon $\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational...