Search found 181 matches
- Thu Jun 22, 2017 5:42 am
- Forum: Number Theory
- Topic: Divisibility... with x,y
- Replies: 1
- Views: 2302
Re: Divisibility... with x,y
The left hand side is even, so $ x^4+y^2$ is even, and $ x, y$ has the same parity. However, if $ x, y$ are odd, we have that $ 4$ divides $ 7^x-3^y$, but $ 4$ doesn't divide $ x^4+y^2$. So $ x, y$ are even, and let's say $ x=2x_1,y=2y_1$. You'll get $ (7^{x_1}-3^{y_1})(7^{x_1}+3^{y_1})$ divides $ 4...
- Thu Jun 22, 2017 4:53 am
- Forum: Geometry
- Topic: IMO 2009 SL(G-1)
- Replies: 1
- Views: 8703
Re: IMO 2009 SL(G-1)
Ugly problem tbh. Let us assign $\angle BAC = 4a$. Then quick angle chases give us, $\angle CAK = a $, $\angle DCI = 45-a $, $\angle CEK = 3a $. We can apply sine law on $\triangle IEK $ and $\triangle CEK $ to get $ \frac{IK}{KC} = \frac{sin(45)*sin (45-a)}{sin(3a)*sin(90-2a)}$ ........(relation 1)...
- Wed Jun 21, 2017 5:26 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 45308
Re: Beginner's Marathon
Problem $23$
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.
- Tue Jun 20, 2017 10:39 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 45308
Re: Beginner's Marathon
They start, from Mirpur, one with the bike($ A$) and the other($ B$) on foot, and from Lalmatia($ C$) on foot of course. After one hour $ A$ will give the bike to $ C$, and $ B$ will stop, and wait for the bike. $ A$ will reach Lamatia after 2 hours. $ C$ rides the bike until he meets $ B$ , and giv...
- Thu Jun 15, 2017 2:47 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 45308
Re: Beginner's Marathon
I edited the post.
- Thu Jun 15, 2017 2:24 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 45308
Re: Beginner's Marathon
Problem $22$ The distance between Mirpur and Lalmatia is $24$ km. Two of three friends need to reach Lalmatia from Mirpur and another friend wants to reach Mirpur from Lalmatia. They have only one bike, which is initially in Mirpur. Each guy may go on foot (with velocity at most $6$ kmph) or on a bi...
- Wed Jun 07, 2017 3:19 pm
- Forum: Junior Level
- Topic: Beginner's Marathon
- Replies: 68
- Views: 45308
Re: Beginner's Marathon
OKAY PEOPLE!! Time to make this thread hot again. Problem $18$ You and your spouse invited $10$ couples to a party. In that party, some of them shook hands with others. But no one shook hands with their spouse. After the party, you asked everyone(including your wife) how many person they shook hands...
- Wed Jun 07, 2017 2:37 pm
- Forum: Divisional Math Olympiad
- Topic: 2017 Regional no.9 Dhaka
- Replies: 11
- Views: 16846
Re: 2017 Regional no.9 Dhaka
No, there are solutions. You have to consider the case where $k+4-4m^2=0$ or $n^2+k-m^2=0$.
- Mon May 01, 2017 1:37 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 194477
Re: Geometry Marathon : Season 3
Problem 42 Let $ABC$ be a triangle with altitudes $AD,BE,CF$ with $D,E,F$ are on sides $BC,CA,AB$, resp. Let $O$ be the circumcenter of triangle $ABC$. $P,Q$ are respectively on $DE,DF$ such that $FP\perp AC$ and $EQ\perp AB$. a) Prove that the midpoint $K$ of $AO$ is circumcenter of triangle $DPQ$...
- Mon May 01, 2017 12:58 pm
- Forum: Geometry
- Topic: Geometry Marathon : Season 3
- Replies: 146
- Views: 194477
Re: Geometry Marathon : Season 3
Solution to Problem 41 Claim: Let $IM$ intersect the perpendicular from $A$ to $BC$ at $G$. $GD \parallel AI$. Proof: As $AG \parallel ID$, it would suffice to prove that $AG=ID$. Hence yielding $AGDI$ a parallelogram. We prove $AG=ID$ by barycentric coordinates. Plugging in points $M=(0,1/2,1/2)$ ...