# Pre RMO Previous Year 2014 Question Paper With Solutions

In this post, you will get **Pre RMO Previous Year 2014 Question Paper With Solutions**.

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**Pre RMO Previous Year 2014 Question Paper Question No 1:**

**A natural number k is such that k ^{2} < 2014 < (k + 1)^{2}. What is the largest prime factor of k?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 1:**

**Pre RMO Previous Year 2014 Question Paper Question No 2:**

**The first term of a sequence is 2014. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2014 ^{th} term of the sequence?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 2:**

**Pre RMO Previous Year 2014 Question Paper Question No 3:**

**Let ABCD be a convex quadrilateral with perpendicular diagonals. If AB = 20, BC = 70 and CD = 90, then what is the value of DA?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 3:**

**Pre RMO Previous Year 2014 Question Paper Question No 4:**

**In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 4:**

**Pre RMO Previous Year 2014 Question Paper Question No 5:**

**If real numbers a, b, c, d, e satisfy a + 1 = 2 + b = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3. What is the value of a ^{2} + b^{2} + c^{2} + d^{2} + e^{2}?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 5:**

**Pre RMO Previous Year 2014 Question Paper Question No 6:**

**What is the smallest possible natural number n for which the equation x ^{2} – nx + 2014 = 0 has integer roots?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 6:**

**Pre RMO Previous Year 2014 Question Paper Question No 7:**

**If , what is the value of ?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 7:**

**Pre RMO Previous Year 2014 Question Paper Question No 8:**

**Let S be a set of real numbers with mean M. If the means of the sets S **U** {15} and S **U** {15,1} are M + 2 and M + 1, respectively, then how many elements does S have?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 8:**

**Pre RMO Previous Year 2014 Question Paper Question No 9:**

**Natural numbers k, I, p and q are such that if a and b are roots of x ^{2} – kx + l = 0 then a + 1/b and b + 1/a are the roots x^{2} – px + q = 0. What is the sum of all possible values of q?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 9:**

**Pre RMO Previous Year 2014 Question Paper Question No 10:**

**In a triangle ABC, X and Y are points on the segments AB and AC, respectively, such that AX : XB = 1 : 2 and AY : YC = 2 : 1. If the area of triangle AXY is 10 then what is the area of triangle ABC?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 10:**

**Pre RMO Previous Year 2014 Question Paper Question No 11:**

**For natural numbers x and y, let (x, y) denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with x < y satisfy the equation xy = x + y + (x, y)?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 11:**

**Pre RMO Previous Year 2014 Question Paper Question No 12:**

**Let ABCD be a convex quadrilateral with angle DAB = angle BDC = 90 degrees. Let the incircles of triangles ABD and BCD touch DB at P and Q, respectively, with P lying in between B and Q. If AD = 999 and PQ = 200 then what is the sum of the radii of the incircles of triangle ABD and BDC?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 12:**

**Pre RMO Previous Year 2014 Question Paper Question No 13:**

**For how many natural numbers n between 1 and 2014 (both inclusive) is an integer?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 13:**

**Pre RMO Previous Year 2014 Question Paper Question No 14:**

**One morning, each member of Manjul’s family drank and 8-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank 1/7-th of the total amount of milk and 2/7-th of the total amount of coffee. How many people are there in Manjul’s family?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 14:**

**Pre RMO Previous Year 2014 Question Paper Question No 15:**

**Let XOY be a triangle with angle XOY = 90 degrees. Let M and N be the midpoints of legs OX and OY, respectively. Suppose that XN = 19 and YM = 22. What is XY?**

**Pre RMO Previous Year 2014 Question Paper with Solutions of Question No 15:**

**Pre RMO Previous Year 2014 Question Paper Question No 16:**

**In a triangle ABC, let I denote the inceter. Let the lines AI, BI and CI intersect the incircle at P, Q and R, respectively. If angle BAC = 40 degrees, what is value of angle QPR in degrees?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 16:**

**Pre RMO Previous Year 2014 Question Paper Question No 17:**

**For a natural number b, let N(b) denote of natural numbers a for which the equation x ^{2} + ax + b = 0 has integer roots. What is the smallest value of b for which N(b) = 20?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 17:**

**Pre RMO Previous Year 2014 Question Paper Question No 18:**

**Let f be a one-to-one function from the set of natural numbers to itself such that f(mn) = f(m) f(n) for all natural numbers m and n. What is the least possible value of f(999)?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 18:**

**Pre RMO Previous Year 2014 Question Paper Question No 19:**

**Let x _{1}, x_{2}, x_{3}, …, x_{2014} be real numbers different from 1 such that x_{1} + x_{2 }+ x_{3 }+ …+ x_{2014} = 1 and . What is the value of ?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 19:**

**Pre RMO Previous Year 2014 Question Paper Question No 20:**

**What is the number of ordered pairs (A, B) where A and B are subsets of {1,2,….,5) such that neither A B not B A?**

**Pre RMO Previous Year 2014 Question Paper Solution of Question No 20:**