IMOmath

General Practice Test

1. (7 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).

2. (37 p.)
How many sequences of 0s and 1s of length 15 are there such that among any three consecutive terms of the sequence there is at most one digit 1?

3. (18 p.)
Real numbers \( x,y,z \) are real numbers greater than 1 and \( w \) is a positive real number. If \( \log_xw=24 \), \( \log_yw=40 \) and \( \log_{xyz}w=12 \), find \( \log_zw \).

4. (18 p.)
Assume that \( A \) is a 40-element subset of \( \{1,2,3,\dots,50\} \), and let \( n \) be the sum of the elements of \( A \). Find the number of possible values of \( n \).

5. (18 p.)
Let \( ABCD \) be a convex quadrilateral such that \( AB\perp BC \), \( AC\perp CD \), \( AB=18 \), \( BC=21 \), \( CD=14 \). Find the perimeter of \( ABCD \).





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