MathDB

4

Part of 2014 Saint Petersburg Mathematical Olympiad

Problems(3)

Another geometry

Source: St Petersburg Olympiad 2014, Grade 11, P4

10/24/2017
Points B1,C1B_1,C_1 are on ACAC and ABAB and B1C1∥BCB_1C_1 \parallel BC. Circumcircle of ABB1ABB_1 intersect CC1CC_1 at LL. Circumcircle CLB1CLB_1 is tangent to ALAL. Prove AL≤AC+AC12AL \leq \frac{AC+AC_1}{2}
geometrycircumcircle
Interesting inequality

Source: St Petersburg Olympiad 2014, Grade 10, P4

10/26/2017
a1≥a2≥...≥a100n>0a_1\geq a_2\geq... \geq a_{100n}>0 If we take from (a1,a2,...,a100n)(a_1,a_2,...,a_{100n}) some 2n+12n+1 numbers b1≥b2≥...≥b2n+1b_1\geq b_2 \geq ... \geq b_{2n+1} then b1+...+bn>bn+1+...b2n+1b_1+...+b_n > b_{n+1}+...b_{2n+1} Prove, that (n+1)(a1+...+an)>an+1+an+2+...+a100n(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}
algebrainequalities
Venerable numbers

Source: St Petersburg Olympiad 2014, Grade 9, P4

10/27/2017
We call a natural number venerable if the sum of all its divisors, including 11, but not including the number itself, is 11 less than this number. Find all the venerable numbers, some exact degree of which is also venerable.
number theory