Almost Mersenne Primes | RMO 2017 Problem 2 | Goa Part 1

RMO 2017 Problem 3 - Roots of a Polynomial

Here is a video post that discusses the roots of a polynomial problem from RMO 2017 problem 3. Watch, learn and enjoy the video.

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RMO 2017 Goa and Maharashtra

Let's solve the Regional Mathematics Olympiad Problem, RMO 2017 from Goa and Maharashtra. Try the problems and check your solutions here.

(\ 1).((\ 16) marks)Consider a chessboard of size (\ 8) units(\ \times8) units (i.e., each small square on the board has a side length of (\ 1) unit).Let (\ S) be the set of all the (\ 81) vertices of all the squares on the board. What is the number of line segments whose vertices are in (\ S), and whose length is a positive integer. (The segments need not be parallel to the sides of the board.)

(\ 2).((\ 16) marks)For any positive integer (\ n), let (\ d(n)) denotes the number of positive divisors of (\ n); and let (\ \phi(n)) denotes the number of elements from the set (\ {1,2,...,n}) that are co-prime to (\ n).(For example (\ d(12)=6 ) and (\ \phi(12)=4).)

Find the smallest positive integer (\ n) such that (\ d(\phi(n))=2017).

(\ 3).((\ 16) marks)Let (\ P(x)) and (\ Q(x)) be polynomials of  degree (\ 6) and degree (\ 3) respectively,such that:
(\ P(x)>Q(x)^2+Q(x)+x^2-6), for all (\ x\in\mathbb{R}).

If all the roots of (\ P(x)) are real numbers, then prove that there exist two roots of (\ P(x)), say (\ \alpha,\beta), such that (\ |\alpha-\beta|<1).

(\ 4).((\ 16) marks)Let (\ l_1,l_2,l_3,\dots,l_{40}) be forty parallel lines.As shown in the diagram, let m be another line that intersects the line (\ l_1) to (\ l_{40}) in the points (\ A_1,A_2.A_3,\dots,A_{40}) respectively.Similarly let n be another line that intersects the lines (\ l_1) to (\ l_{40}) in the points (\ B_1,B_2,B_3,\dots,B_{40}) respectively.

Given that (\ A_1B_1=1), (\ A_{40}B_{40}=14), and the areas of the (\ 39) trapeziums (\ A_1B_1B_2A_2),(\ A_2B_2B_3A_3,\dots),(\ A_{39}B_{39}B_{40}A_{40}) are all equal; then count the number of segments (\ A_iB_i) whose length is a positive integer; where (\ i\in{1,2,\dots,40}).

(\ 5).((\ 18) marks)If (\ a,b,c,d\in\mathbb{R}) such that (\ a>b>c>d>0)  and (\ a+d=b+c);

then prove that :

$$\frac{(a+b)-(c+d)}{\sqrt{2}}>\sqrt{a^2+b^2}-\sqrt{c^2+d^2}$$

(\ 6).((\ 18) marks)Let (\ \triangle{ABC}) be acute-angled; and let (\ \Gamma) be its circumcircle.Let (\ D) be a point on minor arc (\ BC) of (\ \Gamma).Let (\ E) and (\ F) be points on line (\ AD) and (\ AC) respectively, such that (\ BE\perp AD) and (\ DF\perp AC).Prove that (\ EF\parallel BC) if and only if (\ D) is the midpoint of (\ BC).

Regional Math Olympiad 2017

Here are the questions asked in Regional Math Olympiad 2017 and their solutions. Try to solve it first and then see the solutions.

Looking for just the problems? Download the PDF here.

RMO 2017, Problem 1:

Let AOB be a given angle less than \( 180^o \) and let P be an interior point of the angular region determined by \( \angle AOB \) . Show, with proof, how to construct, using only ruler and compass, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB and CP:PD = 1:2.

RMO 2017, Problem 2:

Show that the equation $$ a^3 + (a+1)^3 + (a+2)^3 + (a+3)^3 + (a+4)^3 + (a+5)^3 + (a+6)^3 = b^4 + (b+1)^4 $$ has no solutions in integer a, b.

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RMO 2017, Problem 3:

Let P(x)=x2+12x+b and Q(x)=x2+cx+d be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all real roots of P(Q(x)) = 0

RMO 2017, Problem 4:

Consider n2 unit squares in the xy-plane centered at the point (i, j) with integer coordinates, 1≤i≤n , 1≤j≤n . It is required to color each unit square in such a way that whenever 1≤ i < j and 1 ≤ k < l ≤ n the three squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?

RMO 2017, Problem 5:

Let Ω be a circle with a chord AB which is not a diameter. Let Γ1 be a circle on one side of AB such that it is tangent to AB at C and internally tangent to Ω at D. Likewise let Γ2 be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to Ω at F. Suppose the line DC intersects Ω at X≠D and the line FE intersects Ω at Y≠F. Prove that XY is a diameter of Ω

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RMO 2017 Problem 5

RMO 2017, Problem 6:

Let x, y, z be real numbers, each greater than 1. Prove that

$\frac{x+1}{y+1}$ + $\frac{y+1}{z+1}$ + $\frac{z+1}{x+1}$ $\leq$ $\frac{x-1}{y-1}$ + $\frac{y-1}{z-1}$ + $\frac{z-1}{x-1}$.

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~ Discussion by Souvik Mondal & Writabrata Bhattacharya (Associate Faculty - Cheenta)

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