PRMO and RMO Weekly Challenge (Week -1)

                Good news

Week 2 has arrived 

week 2 questions        

                 MATHFREAKS

test my knowledge of algebra

test my knowledge of geometry

                                         MIOTP (Week-1)

solutions

                                 LEVEL 1

1.a) Find integer n such that both n − 124 and n + 124 are perfect square.

 b) Find an integer a such that 119+a is a perfect square and 118a+1 is a perfect cube

2)what is the probability of constructing a scalene by joining the vertices of a 19 sided regular polygon.

3. How many pairs of   natural number x,y exists for which

such that x,y cannot be both odd or even

4)In triangle ABC, AB is the longest side. Prove that for any point P in the interior

of the triangle, PA+PB > PC.

5)Let p(x) be a polynomial with rational coefficients. Prove that there exists a positive integer n such that the polynomial q(x) defined by

                                  q(x) = p(x + n) − p(x)

has integer coefficients.

6)Given triangle ABC of area 1. Let BM be the perpendicular from B to the bisector

of ∠C. Determine the area of triangle triangle AMC

7)A triangle with integer side has a perimeter of 126 cm find the maximum possible area of the triangle

                                          LEVEL-2

8)There exists a point P inside an equilateral triangle ABC such that PA = 3,PB = 4, and PC = 5. Find the side length of the equilateral triangle.

9)Find the value of x for which

Where [*] represents Greatest Integer function

 10) a) Parallelogram ABCD is given. Prove that the quantity

AX² + CX² − BX² − DX²

does not depend on the choice of point X.

b) Quadrilateral ABCD is not a parallelogram. Prove that all points X that satisfy the relation AX² + CX² − BX² − DX² lie on one line perpendicular to the segment that connects the midpoints of the diagonals.

11)In △ABC bisector AD and midline A1C1 are drawn. They intersect at K. Prove that 2A1K = |b − c|.

12)Let k be a circle with radius r and AB a chord of k such that AB > r. Furthermore, let S be the point on the chord AB satisfying AS = r. The perpendicular bisector of BS intersects k in the points C and D. The line through D and S intersects k for a second time in point E.

Show that the triangle CSE is equilateral.

Solution to these problems will be published next week along with new questions for MIOTP Week 2

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