IMOmath

General Practice Test

1. (5 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).

2. (23 p.)
Let \( a \), \( b \), \( c \), \( d \) be the roots of \( x^4 - x^3 - x^2 - 1 = 0 \). Find \( p(a) + p(b) + p(c) + p(d) \), where \( p(x) = x^6 - x^5 - x^3 - x^2 - x \).

3. (18 p.)
The number \[ \frac1{2\sqrt1+1\sqrt 2}+\frac1{3\sqrt2+2\sqrt3}+\frac1{4\sqrt3+3\sqrt4} + \dots + \frac1{100\sqrt{99}+99\sqrt{100}}\] is a rational number. If it is expressed as \( \frac pq \) for two relatively prime integers \( p \) and \( q \) evaluate \( p+q \).

4. (28 p.)
Let \( ABC \) be a rectangular triangle such that \( \angle C=90^o \) and \( AC = 7 \), \( BC = 24 \). Let \( M \) be the midpoint of \( AB \) and \( D \) point on the same side of \( AB \) as \( C \) such that \( DA = DB = 15 \). The area of the triangle \( CDM \) can be expressed as \( \frac{p\sqrt q}r \) for positive integers \( p \), \( q \), \( r \) such that \( q \) is not divisible by a perfect square and \( (p,r)=1 \). Find area \( p+q+r \).

5. (23 p.)
Let \( ABC \) be a triangle with sides 3, 4, 5 and \( DEFG \) a \( 6 \times 7 \) rectangle. A line divides \( \triangle ABC \) into a triangle \( T_1 \) and a trapezoid \( R_1 \). Another line divides the rectangle \( DEFG \) into a triangle \( T_2 \) and a trapezoid \( R_2 \), in such a way \( T_1\sim T_2 \) and \( R_1\sim R_2 \). The smallest possible value for the area of \( T_1 \) can be expressed as \( p/q \) for two relatively prime positive integers \( p \) and \( q \). Evaluate \( p+q \).





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