Log In
Register
IMOmath
Olympiads
Book
Training
Forum
IMOmath
General Practice Test
1.
(23 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?
2.
(21 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.
(16 p.)
Given a convex polyhedron with 26 vertices, 60 edges and 36 faces, 24 of the faces are triangular and 12 are quadrilaterals. A space diagonal is a line segment connecting two vertices which do not belong to the same face. How many space diagonals does the polyhedron have?
4.
(30 p.)
If \( ABCD \) is a convex quadrilateral with \( AB=200 \), \( BC=153 \), \( BD=300 \), \( \angle BAC=\angle BDC<90^{\circ} \) and \( \angle ABD=\angle BCD \), determine \( CD. \)
5.
(7 p.)
Let \( \alpha \) be the angle between vectors \( \vec a \) and \( \vec b \) with \( \vec a=2 \) and \( \vec b=3 \), given that the vectors \( \vec m=2\vec a\vec b \) and \( \vec n=\vec a+5\vec b \) are orthogonal. If \( \cos\alpha=\frac pq \) with \( q>0 \) and \( \gcd(p,q)=1 \), compute \( p+q \).
20052020
IMOmath.com
 imomath"at"gmail.com  Math rendered by
MathJax
Home

Olympiads

Book

Training

IMO Results

Forum

Links

About

Contact us