MathDB

Minimum value of expression with prime

Source: Moldova TST Problem 3

3/31/2015

Let

p

be a fixed odd prime. Find the minimum positive value of

E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}

where

x,y \in \mathbb{Z}_{+}

.

number theory

Concurrency of lines involving altitudes

Source: Moldova TST Problem 7

4/1/2015

Consider an acute triangle

ABC

, points

E,F

are the feet of the perpendiculars from

B

and

C

in

\triangle ABC

. Points

I

and

J

are the projections of points

F,E

on the line

BC

, points

K,L

are on sides

AB,AC

respectively such that

IK \parallel AC

and

JL \parallel AB

. Prove that the lines

IE

,

JF

,

KL

are concurrent.

geometry

a cute geometric inequality

Source: Moldova TST Problem 3, day 3

3/31/2015

The tangents to the inscribed circle of

\triangle ABC

, which are parallel to the sides of the triangle and do not coincide with them, intersect the sides of the triangle in the points

M,N,P,Q,R,S

such that

M,S\in (AB)

,

N,P\in (AC)

,

Q,R\in (BC)

. The interior angle bisectors of

\triangle AMN

,

\triangle BSR

and

\triangle CPQ

, from points

A,B

and respectively

C

have lengths

l_{1}

,

l_{2}

and

l_{3}

.\\ Prove the inequality:

\frac {1}{l^{2}_{1}}+\frac {1}{l^{2}_{2}}+\frac {1}{l^{2}_{3}} \ge \frac{81}{p^{2}}

where

p

is the semiperimeter of

\triangle ABC

.

inequalitiesgeometric inequality

3

Problems(3)