## Search found 86 matches

Fri Feb 10, 2017 9:27 pm
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
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
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
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

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

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
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
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...