MathDB

Problem 4

Part of 2003 Federal Math Competition of S&M

Problems(3)

locus of tangency points as center moves

Source: Serbia 2003 1st Grade P4

5/12/2021
An acute angle with the vertex OO and the rays Op1Op_1 and Op2Op_2 is given in a plane. Let k1k_1 be a circle with the center on Op1Op_1 which is tangent to Op2Op_2. Let k2k_2 be the circle that is tangent to both rays Op1Op_1 and Op2Op_2 and to the circle k1k_1 from outside. Find the locus of tangency points of k1k_1 and k2k_2 when center of k1k_1 moves along the ray Op1Op_1.
geometryLocus
nice problem

Source: Serbia and Montenegro 2003

7/21/2006
Let SS be the subset of NN(NN is the set of all natural numbers) satisfying: i)Among each 20032003 consecutive natural numbers there exist at least one contained in SS; ii)If n∈Sn \in S and n>1n>1 then [n2]∈S[\frac{n}{2}] \in S Prove that:S=NS=N I hope it hasn't posted before. :lol: :lol:
number theory proposednumber theory
subset partition

Source: Serbia 2003

4/28/2008
Let n n be an even number, and S S be the set of all arrays of length n n whose elements are from the set {0,1} \left\{0,1\right\}. Prove that S S can be partitioned into disjoint three-element subsets such that for each three arrays \left(a_i\right)_{i \equal{} 1}^n, \left(b_i\right)_{i \equal{} 1}^n, \left(c_i\right)_{i \equal{} 1}^n which belong to the same subset and for each i∈{1,2,...,n} i\in\left\{1,2,...,n\right\}, the number a_i \plus{} b_i \plus{} c_i is divisible by 2 2.
inductionconicsellipsevectorcombinatorics proposedcombinatorics