MathDB

7

Part of 1996 IMO Shortlist

Problems(3)

f(x + 13/42) + f(x) = f(x + 1/6) + f(x + 1/7)

Source: IMO Shortlist 1996, A7

8/9/2008
Let f f be a function from the set of real numbers R \mathbb{R} into itself such for all x∈R, x \in \mathbb{R}, we have ∣f(x)∣≤1 |f(x)| \leq 1 and f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right). Prove that f f is a periodic function (that is, there exists a non-zero real number c c such f(x\plus{}c) \equal{} f(x) for all x∈R x \in \mathbb{R}).
functionalgebrapolynomialfunctional equationIMO Shortlistperiodic function
Geometric Inequality

Source: IMO Shortlist 1996

9/29/2006
Let ABCABC be an acute triangle with circumcenter OO and circumradius RR. AOAO meets the circumcircle of BOCBOC at A′A', BOBO meets the circumcircle of COACOA at B′B' and COCO meets the circumcircle of AOBAOB at C′C'. Prove that OA′⋅OB′⋅OC′≥8R3.OA'\cdot OB'\cdot OC'\geq 8R^{3}. Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
inequalitiesgeometrycircumcircletrigonometrytrig identitiesIMO ShortlistTriangle
two injective surjective function

Source: ISL 1996, C7

6/6/2005
let V V be a finitive set and g g and f f be two injective surjective functions from V VtoV V.let T T and S S be two sets such that they are defined as following" S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\} T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\} we know that S \cup T \equal{} V, prove: for each w∈V:f(w)∈S w \in V : f(w) \in S if and only if g(w)∈S g(w) \in S
functionsymmetrycombinatoricspartitionIMO Shortlist