Search found 86 matches
- Fri Dec 09, 2016 9:30 am
- Forum: Secondary Level
- Topic: The Largest Integer!
- Replies: 0
- Views: 1944
The Largest Integer!
We say a non-negative integer $n$ "contains" another non-negative integer $m$, if the digits of its decimal expansion appear consecutively in the decimal expansion of $n$. For example, $2016$ contains $2$, $0$, $1$, $6$, $20$, $16$, $201$, and $2016$. Find the largest integer $n$ that does not conta...
- Sat Aug 27, 2016 8:07 pm
- Forum: Number Theory
- Topic: India TST 2014
- Replies: 3
- Views: 3569
Re: India TST 2014
Yeah , that's easy to see after some experiment. But u have to show your whole own solution. Hint: Use parity,divisibility (may be useful fact, $x$ & $y$ divides both sides),etc. And split it into cases. It's easy to solve it. So I don't think to show whole solution. It can be solved easily by basi...
- Sat Aug 27, 2016 12:00 pm
- Forum: Number Theory
- Topic: India TST 2014
- Replies: 3
- Views: 3569
- Fri Aug 26, 2016 9:14 pm
- Forum: Number Theory
- Topic: USAJMO 2016
- Replies: 1
- Views: 2825
USAJMO 2016
Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.
This problem was proposed by Evan Chen.
This problem was proposed by Evan Chen.
- Thu Aug 18, 2016 11:40 pm
- Forum: Number Theory
- Topic: Perfect Cube
- Replies: 1
- Views: 2516
Perfect Cube
If $ \frac{a }{b}+ \frac{b}{c}+ \frac{c}{a}$ is integer.
Show that $ abc$ is perfect cube.
Show that $ abc$ is perfect cube.
- Tue Aug 09, 2016 9:22 pm
- Forum: Geometry
- Topic: A Problem of Romanian TST
- Replies: 10
- Views: 7994
Re: A Problem of Romanian TST
Then tell your whole solution.Raiyan Jamil wrote:#kazi zareer... none of the three quads you've mentioned are cyclic quads... :/
- Mon Aug 08, 2016 9:32 am
- Forum: Geometry
- Topic: Clash of circles!
- Replies: 0
- Views: 2103
Clash of circles!
In $\triangle ABC$ Let $W_a$ be the circle that passes throw $B,C $ and it is tangent to the incircle of $\triangle ABC$. We define $W_b,W_c$ similarly. Let $W_a\cap W_b=B', W_a\cap W_c=C'$ let $BC'\cap AC=B'', CB'\cap AB=C''$ let $I$ be the incenter of $\triangle ABC$. Prove that $B''$,$I$, $C''$ a...
- Mon Aug 08, 2016 1:31 am
- Forum: International Mathematical Olympiad (IMO)
- Topic: 2010 C3
- Replies: 0
- Views: 2258
2010 C3
$2500$ chess kings have to be placed on a $100 \times 100$ chessboard so that (i) no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); (ii) each row and each column contains exactly $25$ kings. Find the number of such arrangements. (Two arrangement...
- Sat Aug 06, 2016 8:22 pm
- Forum: National Math Olympiad (BdMO)
- Topic: A Acute angled triangle
- Replies: 1
- Views: 1912
A Acute angled triangle
$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE $ and $CF$. The line parallel to $ DF $ through $E$ meets $BC$ at $Y$ and $BA$ at $X.$ $DF$ and $CA $ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S.$ Given, $\angle B=33 $ d...
- Fri Aug 05, 2016 1:21 pm
- Forum: Geometry
- Topic: Circumcircle is tangent to the circumcircle
- Replies: 2
- Views: 2775
Circumcircle is tangent to the circumcircle
In $\triangle ABC$,$AB>AC$,let $H$ be $\triangle ABC$'s orthocenter,$M$ be $BC$'s midpoint.point $S$ is on $BC$ satisfies $\angle BHM=\angle CHS$.Point $P$ is on $HS$ so that $AP \perp HS$
Prove that the circumcircle of $\triangle MPS$ is tangent to the circumcircle of $\triangle ABC$
Prove that the circumcircle of $\triangle MPS$ is tangent to the circumcircle of $\triangle ABC$