Olympiad Questions
Mathematics Initiatives in Nepal
1 Practice Questions
1. If p1 , p2 , p3 , . . . is an arithmetic sequence with common difference 1 and
p1 +p2 +p3 +· · ·+p98 = 137, what is the value of p2 +p4 +p6 +· · ·+p98 ?
√
2. Find the sum of the integer solutions to the equation 4 n = 7−12√ 4 n.
3. Find all 3 natural numbers x, y, z satisfying y · xz = z · xy + 10.
4. Find the value of
1 1 1
+ +
1 + a + ab 1 + b + bc 1 + c + ca
if a, b, and c are reals such that abc = 1.
5. Four friends bought a ball. The first one paid half of the ball price.
The second one gave one-third of the money that the other three gave.
The third one paid a quarter of the sum paid by the other three. The
fourth paid $5. How much did the ball cost?
6. Find the smallest positive integer n ≥ 10 such that n + 6 is a prime
and 9n + 7 is a perfect square.
7. Given that x, y, and z are positive real numbers, let
√ √ √
M = x + 2 + y + 5 + z + 10
√ √ √
N = x+1+ y+1+ z+1
Find the minimum possible value of M 2 − N 2 .
1
8. The expressions A = 1 × 2 + 3 × 4 + 5 × 6 + · · · + 37 × 38 + 39 and
B = 1 + 2 × 3 + 4 × 5 + · · · + 36 × 37 + 38 × 39 are obtained by writing
multiplication and addition operators in an alternating pattern between
successive integers. Find the positive difference between integers A and
B.
9. There is a prime number p such that 16p + 1 is the cube of a positive
integer. Find p.
10. Let a and b be positive integers satisfying
ab + 1 3
< .
a+b 2
The maximum possible value of
a3 b3 + 1
a3 + b3
is pq , where p and q are relatively prime positive integers. Find p + q.
11. The circle ω touches the circle Ω internally at P . The center 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 P Y intersect ω at Z. If Y Z = 2P Z, what is
the magnitude of ∠P Y X in degrees?
