MathDB

1

Part of 2017 Junior Balkan Team Selection Tests - Romania

Problems(4)

orthocenter wanted, reflections of interior point lie on circumcircle of acute

Source: 2017 Romania JBMO TST1 p1

5/16/2020
Let PP be a point in the interior of the acute-angled triangle ABCABC. Prove that if the reflections of PP with respect to the sides of the triangle lie on the circumcircle of the triangle, then PP is the orthocenter of ABCABC.
geometrycircumcirclereflectionorthocenter
Number Theory, Proving primes for divisors of a number

Source: JBMO TST 3 2017 P1

11/14/2018
Let nn and kk be two positive integers such that 1≤n≤k1\leq n \leq k. Prove that, if dk+kd^k+k is a prime number for each positive divisor dd of nn, then n+kn+k is a prime number.
number theoryprime numbers
2 player game on a 2xn rectangular grid

Source: 2017 Romania JBMO TST 4.1 - Estonian Olympiad, 2009

6/27/2020
Alina and Bogdan play a game on a 2\times n rectangular grid (n≥2n\ge 2) whose sides of length 22 are glued together to form a cylinder. Alternating moves, each player cuts out a unit square of the grid. A player loses if his/her move causes the grid to lose circular connection (two unit squares that only touch at a corner are considered to be disconnected). Suppose Alina makes the first move. Which player has a winning strategy?
combinatoricsgamewinning strategygame strategy
a^2 + b^2 + c^2 >= 3(a + b + c) if a, b, c \in [-1, 1] ,a + b + c + abc = 0

Source: 2017 Romania JBMO TST 5.1

6/27/2020
If a,b,c∈[−1,1]a, b, c \in [-1, 1] satisfy a+b+c+abc=0a + b + c + abc = 0, prove that a2+b2+c2≥3(a+b+c)a^2 + b^2 + c^2 \ge 3(a + b + c) . When does the equality hold?
inequalitiesalgebra