PRMO - 2015 A - Questions, Discussions, Hints, Solutions
This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Problems
A man walks a certain distance and rides back in 334 hours; he could ride both ways in 212 hours. How many hours would it take him to walk both ways? Time and Distance
Positive integers a and b are such that a+b=a/b+b/a. What is the value of a2+b2 ? Algebra
The equations x2−4x+k=0 and x2+kx−4=0, where k is a real number, have exactly one common root. What is the value of k? Quadratic Equation
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)? PolynomialEquation
How many line segments have both their endpoints located at the vertices of a given cube? Combinatorics
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
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
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
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
What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ? Geometry
In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that ∠BPC = 90◦ . If r1,r2,r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value of r1+r2+r3? Geometry
Let a, b, and c be real numbers such that a−7b+8c=4 and 8a+4b−c=7. What is the value of a2−b2+c2 ? Algebra
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
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
If 3x+2y=985 and 3x−2y=473, what is the value of xy? Algebra
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 ∠BAE=40◦.If∠DAE=∠DFE, what is the magnitude of ∠ADF in degrees? Geometry
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
Let a, b and c be such that a+b+c=0 and P=a22a2+bc+ b22b2+ca + c22c2+ab is defined. What is the value of P? Algebra
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 PY>PX. Let PY intersect ω at Z. If YZ=2PZ, what is the magnitude of ∠PYX in degrees? Geometry
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
This is a work in progress. Please post your answers in the comment. We will update them here. Point out any error that you see here. Thank you.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Problems
A man walks a certain distance and rides back in 334 hours; he could ride both ways in 212 hours. How many hours would it take him to walk both ways? Time and Distance
Positive integers a and b are such that a+b=a/b+b/a. What is the value of a2+b2 ? Algebra
The equations x2−4x+k=0 and x2+kx−4=0, where k is a real number, have exactly one common root. What is the value of k? Quadratic Equation
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)? PolynomialEquation
How many line segments have both their endpoints located at the vertices of a given cube? Combinatorics
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
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
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
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
What is the greatest possible perimeter of a right-angled triangle with integer side lengths if one of the sides has length 12 ? Geometry
In rectangle ABCD, AB = 8 and BC = 20. Let P be a point on AD such that ∠BPC = 90◦ . If r1,r2,r3 are the radii of the incircles of triangles APB, BPC and CPD, what is the value of r1+r2+r3? Geometry
Let a, b, and c be real numbers such that a−7b+8c=4 and 8a+4b−c=7. What is the value of a2−b2+c2 ? Algebra
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
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
If 3x+2y=985 and 3x−2y=473, what is the value of xy? Algebra
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 ∠BAE=40◦.If∠DAE=∠DFE, what is the magnitude of ∠ADF in degrees? Geometry
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
Let a, b and c be such that a+b+c=0 and P=a22a2+bc+ b22b2+ca + c22c2+ab is defined. What is the value of P? Algebra
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 PY>PX. Let PY intersect ω at Z. If YZ=2PZ, what is the magnitude of ∠PYX in degrees? Geometry
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
