IMOmath

General Practice Test

1. (36 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.

2. (19 p.)
If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

3. (2 p.)
The square \( \begin{array}{|c|c|c|} \hline x&20&151 \\\hline 38 & & \\ \hline & & \\ \hline\end{array} \) is magic, i.e. in each cell there is a number so that the sums of each row and column and of the two main diagonals are all equal. Find \( x \).

4. (36 p.)
Let \( K \) and \( L \) be the points on the sides \( AB \) and \( BC \) of an equilateral triangle \( ABC \) such that \( AK=5 \) and \( CL=2 \). If \( M \) is the point on \( AC \) such that \( \angle KML=60^o \), and if the area of the triangle \( KML \) is equal to \( 14\sqrt3 \) then the side of the triangle \( ABC \) can assume two values \( \frac{a\pm \sqrt b}c \) for some natural numbers \( a \), \( b \), and \( c \). If \( b \) is not divisible by a perfect square other than 1, find the value of \( b \).

5. (4 p.)
The set \( A \) consists of \( m \) consecutive integers with sum \( 2m \). The set \( B \) consists of \( 2m \) consecutive integers with sum \( m \). The difference between the largest elements of \( A \) and \( B \) is 99. Find \( m \).





2005-2018 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us