2013 TSTST Problems
Contents
[hide]Day 1
Problem 1
Let be a triangle and
,
,
be the midpoints of arcs
,
,
on the circumcircle. Line
passes through the feet of the perpendiculars from
to
and
. Line
passes through the feet of the perpendiculars from
to
and
. Let
denote the intersection of lines
and
. Define points
and
similarly. Prove that triangle
and
are similar to each other.
Problem 2
A finite sequence of integers is called regular if there exists a real number
satisfying
Given a regular sequence
, for
we say that the term
is forced if the following condition is satisfied: the sequence
is regular if and only if
. Find the maximum possible number of forced terms in a regular sequence with
terms.
Problem 3
Divide the plane into an infinite square grid by drawing all the lines and
for
. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly
of the squares are black and no two black squares are adjacent). Let
and
be odd integers, and let
be a point in the interior of any white square such that
is irrational. Shoot a laser out of this point with slope
; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.
Day 2
Problem 4
Circle , centered at
, is internally tangent to circle
, centered at
, at
. Let
and
be variable points on
and
, respectively, such that line
is tangent to
(at
). Determine the locus of
-- the circumcenter of triangle
.
Problem 5
Let be a prime. Prove that any complete graph with
vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by
.
Problem 6
Let be the set of positive integers. Find all functions
that satisfy the equation
for all
.
(Here
and
for every integer
greater than
.)
Problem 7
A country has cities, labelled
. It wants to build exactly
roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly
, and it is also not permitted to put a road between cities
and
. Let
be the total number of possible ways to build these roads.
(a) For all odd
, prove that
is divisible by
.
(b) For all even
, prove that
is divisible by
.
Problem 8
Define a function by
,
for every positive integer
. Prove that
leave distinct remainders when divided by
.
Problem 9
Let be a rational number in the interval
and let
. Call a subset
of the plane good if
is unchanged upon rotation by
around any point of
(in both clockwise and counterclockwise directions). Determine all values of
satisfying the following property: The midpoint of any two points in a good set also lies in the set.