IMOmath

General Practice Test

1. (11 p.)
Let \( a \) be the coefficient of \( x^2 \) in the polynomial \[ (1-x)(1+2x)(1-3x)\dots (1+14x)(1-15x).\] Determine \( |a| \)

2. (13 p.)
Let \( 0 < a < b < c < d \) be integers such that \( a \), \( b \), \( c \) is an arithmetic progression, \( b \), \( c \), \( d \) is a geometric progression, and \( d - a = 30 \). Find \( a + b + c + d \).

3. (25 p.)
It is given that \( 181^2 \) can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

4. (25 p.)
At the basement of a building with 5 floors, Adam, Bob, Cindy, Diana and Ernest entered the elevator. The elevator goes only up and doesn’t come back, and each person gets out of the elevator at one of the five floors. In how many ways can the five people leave the elevator in such a way that at no time are there a male and a female alone in the elevator?

5. (23 p.)
Assume that all sides of the convex hexagon \( ABCDEF \) are equal and the opposite sides are parallel. Assume further that \( \angle FAB = 120^o \). The \( y \)-coordinates of \( A \) and \( B \) are 0 and 2 respectively, and the \( y \)-coordinates of the other vertices are 4, 6, 8, 10 in some order. The area of \( ABCDEF \) can be written as \( a\sqrt b \) for some integers \( a \) and \( b \) such that \( b \) is not divisible by a perfect square other than 1. Find \( a+b \).





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