7th University of Rochester. Math Olympiad. Saturday, February 16, 2013. 9:30
AM - 12:30 PM. Show all work (each step and computation) to receive full credit.
7th University of Rochester Math Olympiad
Saturday, February 16, 2013 9:30 AM - 12:30 PM Show all work (each step and computation) to receive full credit. You may use back pages if necessary. Calculators are not permitted. The olympiad consists of four problems. Please submit your work for all the problems, even if some pages are blank.
No. 1 2 3 4 TOTAL
VALUE 10 10 10 10 40
REGISTRATION NO. :
1
SCORE
1. Let 4ABC be a triangle with m ∠BAC = 120◦ . Suppose that the bisectors of angles ∠BAC, ∠ABC and ∠ACB meet sides BC, AC and AB at points D, E and F , respectively. Prove that 4DEF is a right triangle.
2
2. Compute, with explanation, the determinant of the following n × n matrix 1 2 3 4 ... n 2 1 2 3 . . . n − 1 3 2 1 2 . . . n − 2 . 4 3 2 1 . . . n − 3 .. .. .. .. .. .. . . . . . . n n − 1 n − 2 n − 3 ... 1
3
3. Let n ≥ 2 be a positive integer and consider n arbitrary functions f1 , f2 , . . . , fn : [0, 1] → R. Prove that there exist x1 , x2 , . . . , xn ∈ [0, 1] such that |f1 (x1 ) + f2 (x2 ) + . . . + fn (xn ) − x1 x2 · · · xn | ≥
4
1 . 2n
4. Define a sequence of real numbers a1 = an = n k
=
n! k!(n−k)!
+ 11 , a2 =
1 1
+ 12 + 11 , and in general
1 1 1 n + n + ··· + n , 0
where
1 1
1
n
as usual. Determine, with proof, the value of lim an . n→∞
5
