MathDB

2

Part of 2013 All-Russian Olympiad

Problems(6)

Orthocentre of ADE is the midpoint of BC

Source: ARMO 2013, 9th grade, p2

5/20/2013
Acute-angled triangle ABCABC is inscribed into circle Ω\Omega. Lines tangent to Ω\Omega at BB and CC intersect at PP. Points DD and EE are on ABAB and ACAC such that PDPD and PEPE are perpendicular to ABAB and ACAC respectively. Prove that the orthocentre of triangle ADEADE is the midpoint of BCBC.
geometrycircumcirclegeometric transformationreflectiontrigonometryangle bisectorgeometry proposed
Ten quadratic trinomials

Source: ARMO 2013, 9th grade, p6

5/20/2013
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
quadraticsalgebrapolynomialarithmetic sequencealgebra proposed
Circle with n points

Source: 2013 All-Russian Olympiad, Final Round, grade 10, Day 1, P:2

5/16/2014
Circle is divided into nn arcs by nn marked points on the circle. After that circle rotate an angle 2πk/n 2\pi k/n (for some positive integer k k ), marked points moved to nn new points , dividing the circle into n n new arcs. Prove that there is a new arc that lies entirely in the one of the old arсs. (It is believed that the endpoints of arcs belong to it.)
I. Mitrophanov
rotationgeometrygeometric transformationinvariantcombinatorics proposedcombinatorics
10 polynomials

Source: 2013 All-Russian Olympiad, Final Round, grade 10, Day 2, P:6

5/16/2014
Peter and Vasil together thought of ten 5-degree polynomials. Then, Vasil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Vasil could have called out?
A. Golovanov
algebrapolynomialalgebra proposed
ARMO; easy pyramid

Source: AllRussian MO-2013, Grade 11, P:2

7/18/2013
The inscribed and exscribed sphere of a triangular pyramid ABCDABCD touch her face BCDBCD at different points XX and YY. Prove that the triangle AXYAXY is obtuse triangle.
geometry3D geometrypyramidspheregeometry proposed
Prove a^2 + b^2 + c^2 + d^2 is at least abcd

Source: ARMO 2013 Grade 11 P6

5/22/2013
Let a,b,c,da,b,c,d be positive real numbers such that 2(a+b+c+d)≥abcd 2(a+b+c+d)\ge abcd . Prove that a2+b2+c2+d2≥abcd. a^2+b^2+c^2+d^2 \ge abcd .
inequalitiesinequalities proposed