## Search found 86 matches

- Fri Feb 10, 2017 9:27 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 National Round Secondary 10
- Replies:
**5** - Views:
**2401**

### BdMO 2017 National Round Secondary 10

$p$ is an odd prime. The integer $k$ is in the range $1 \leq k \leq p-1.$ Let $a_k$ be the number of divisors of $kp + 1 $ that are greater than or equal to $k$ and less than $p.$ Find the value of $a_1 + a_2 + \dots + a_{p-1}.$

- Fri Feb 10, 2017 9:24 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 National Round Secondary 9
- Replies:
**2** - Views:
**1817**

### BdMO 2017 National Round Secondary 9

In a cyclic quadrilateral $ABCD$ with circumcenter $O,$ the lines $BC$ and $AD$ intersect at $E.$ The lines $AB$ and $CD$ intersect at $F.$ A point $P$ satisfying $\angle EPD = \angle FPD = \angle BAD$ is chosen inside of $ABCD.$ The line $FO$ intersects the lines $AD,EP,BC$ at $X,Q,Y$ respectively....

- Fri Feb 10, 2017 9:12 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 National Round Secondary 8
- Replies:
**2** - Views:
**1803**

### BdMO 2017 National Round Secondary 8

The sequence $\left \{ a_n \right \}$ is defined by $a_{n+1} = 2(a_n - a_{n-1}),$ where $a_0 = 1,$ $a_1 = 1$ for all positive integers $n.$ What is the remainder of $a_{2016}$ upon division by $2017$? Provide a proof of your answer.

- Fri Feb 10, 2017 9:06 pm
- Forum: National Math Olympiad (BdMO)
- Topic: BdMO 2017 National Round Secondary 7
- Replies:
**14** - Views:
**4706**

### BdMO 2017 National Round Secondary 7

$100$ pictures of

*BdMO*math campers were painted by*Urmi*. Exactly $k$ colors were used in each picture. There is a common color in every $20$ pictures. But, there is no common color in all $100$ pictures. Find the smallest possible value of $k.$- Sat Feb 04, 2017 8:58 pm
- Forum: Geometry
- Topic: Circle is tangent to circumcircle and incircle
- Replies:
**3** - Views:
**1837**

### Re: Circle is tangent to circumcircle and incircle

Extend $DF$ and it will intersect the bisector of $\angle BAC$ at $Y$.Absur Khan Siam wrote:Will bisector $\angle BAC$ intersect both $DE$ and $DF$?

- Sat Feb 04, 2017 12:48 pm
- Forum: National Math Camp
- Topic: National Camp 2013 Geomretry Prb 2
- Replies:
**1** - Views:
**3390**

### Re: National Camp 2013 Geomretry Prb 2

*https://artofproblemsolving.com/communi ... 84p2641339*

$k = 1 $

The original source of this problem was: Turkish TST 2012 Problem 4

- Sat Feb 04, 2017 12:43 pm
- Forum: Geometry
- Topic: A Geometry Problem, to prove equal angles, will be fun!
- Replies:
**0** - Views:
**1256**

### A Geometry Problem, to prove equal angles, will be fun!

et $ABC$ be a triangle inscribed in circle $(O)$, incenter $I$. Circle $(K)$ touches $CA,AB$ at $E,F$ and touches $(O)$ internally. $AI$ cuts $(O)$ again at $P$. $PQ$ is diameter of $(O)$. $QI$ cuts $BC$ at $D$. $M,N$ are midpoints $DI$ and $KA$. $R$ is on perpendicular bisector of $AQ$ such that $M...

- Sat Feb 04, 2017 12:19 pm
- Forum: Geometry
- Topic: Circle is tangent to circumcircle and incircle
- Replies:
**3** - Views:
**1837**

### Circle is tangent to circumcircle and incircle

In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle...

- Fri Feb 03, 2017 5:26 pm
- Forum: National Math Olympiad (BdMO)
- Topic: How two altitudes determine the third
- Replies:
**3** - Views:
**1391**

### How two altitudes determine the third

If the lengths of two altitudes drawn from two vertices of a triangle on their opposite sides are $2014$ and $1$ unit, then what will be the length of the altitude drawn from the third vertex of the triangle on its opposite side?

Source: BdMO National 2014

Source: BdMO National 2014

- Wed Feb 01, 2017 2:50 pm
- Forum: Combinatorics
- Topic: BAMO P2
- Replies:
**1** - Views:
**1382**

### BAMO P2

A lock has $16$ keys arranged in a $4 \times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions to...