MathDB

2

Part of 2002 France Team Selection Test

Problems(2)

Elements in S such that difference of divisors is 22

Source: French TST 2002

6/17/2011
Consider the set SS of integers kk which are products of four distinct primes. Such an integer k=p1p2p3p4k=p_1p_2p_3p_4 has 1616 positive divisors 1=d1<d2<…<d15<d16=k1=d_1<d_2<\ldots <d_{15}<d_{16}=k. Find all elements of SS less than 20022002 such that d9−d8=22d_9-d_8=22.
number theory proposednumber theory
condition for angle AIO &lt; 90

Source: France TST 2002; CHKMO 1999

6/30/2007
Let ABC ABC be a non-equilateral triangle. Denote by I I the incenter and by O O the circumcenter of the triangle ABC ABC. Prove that ∠AIO≤π2 \angle AIO\leq\frac{\pi}{2} holds if and only if 2\cdot BC\leq AB\plus{}AC.
geometryincentercircumcircletrigonometrygeometry proposed