PRMO - 2015 A - Questions, Discussions, Hints, Solutions

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Problems

1. A man walks a certain distance and rides back in $3 \frac{3}{4}$ hours; he could ride both ways in $2 \frac{1}{2}$ hours. How many hours would it take him to walk both ways?
   Time and Distance
   
2. Positive integers a and b are such that $a + b = a/b + b/a$. What is the value of $a^2 + b^2$ ?
   Algebra
   
3. The equations $x^2 − 4x + k = 0$ and $x^2 + kx − 4 = 0$, where k is a real number, have exactly one common root. What is the value of k?
   Quadratic Equation
   
4. Let P(x) be a non-zero polynomial with integer coefficients. If P(n) is divisible by n for each positive integer n, what is the value of P(0)?
   Polynomial Equation
   
5. How many line segments have both their endpoints located at the vertices of a given cube?
   Combinatorics
   
6. Let E(n) denote the sum of the even digits of n. For example, $E(1243) = 2 + 4 = 6$. What is the value of $E(1) + E(2) + E(3) + · · · + E(100)$?
   Number Theory
   
7. How many two-digit positive integers N have the property that the sum of N and the number obtained by reversing the order of the digits of N is a perfect square?
   Number Theory
   
8. The figure below shows a broken piece of a circular plate made of glass.
   
   
   C is the midpoint of AB, and D is the midpoint of arc AB. Given that AB = 24 cm and CD = 6 cm, what is the radius of the plate in centimeters? (The figure is not drawn to scale.)
   Geometry
   
9. A 2 × 3 rectangle and a 3 × 4 rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
   Geometry
   
10. What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ?
    Geometry
    
11. In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that $\angle BPC$ = 90◦ . If $r_1, r_2, r_3$ are the radii of the incircles of triangles APB, BPC and CPD, what is the value of $r_1 + r_2 + r_3$?
    Geometry
    
12. Let a, b, and c be real numbers such that $a − 7b + 8c = 4$ and $8a + 4b − c = 7$. What is the value of $a^2 − b^2 + c^2$ ?
    Algebra
    
13. Let n be the largest integer that is the product of exactly 3 distinct prime numbers, $x$, $y$ and $10x + y$, where x and y are digits. What is the sum of the digits of n?
    Number Theory
    
14. At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party?
    Combinatorics
    
15. If $3^x + 2^y = 985$ and $3^x − 2^y = 473$, what is the value of $xy$?
    Algebra
    
16. In acute-angled triangle ABC, let D be the foot of the altitude from A, and E be the midpoint of BC. Let F be the midpoint of AC. Suppose $\angle BAE = 40◦ . If \angle DAE = \angle DF E$, what is the magnitude of $\angle ADF$ in degrees?
    Geometry
    
17. A subset B of the set of first 100 positive integers has the property that no two elements of B sum to 125. What is the maximum possible number of elements in B?
    Set Theory
    
18. Let a, b and c be such that $a + b + c = 0$ and
    $P=$ $\frac{a^2}{2a^2+bc}$+ $\frac{b^2}{2b^2 +ca}$ + $\frac{c^2}{2c^2+ab}$
    is defined. What is the value of $P$?
    Algebra
    
19. The circle ω touches the circle Ω internally at P. The centre O of Ω is outside ω. Let $XY$ be a diameter of Ω which is also tangent to ω. Assume $P Y > P X$. Let PY intersect ω at Z. If $YZ = 2PZ$, what is the magnitude of $\angle PYX$ in degrees?
    Geometry
    
20. The digits of a positive integer n are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when n is divided by 37?
    Divisibility
    
    PRMO 2015 Set A (Pre Regional Math Olympiad, India) – Download Question Paper

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