MathDB

4

Part of 2017 Junior Balkan Team Selection Tests - Romania

Problems(5)

rectangle m x n can be tilled with t-shapes and p-shapes

Source: 2017 Romania JBMO TST 1.4

6/27/2020
Two right isosceles triangles of legs equal to 11 are glued together to form either an isosceles triangle - called t-shape - of leg 2\sqrt2, or a parallelogram - called p-shape - of sides 11 and 2\sqrt2. Find all integers mm and n,m,n≥2n, m, n \ge 2, such that a rectangle m×nm \times n can be tilled with t-shapes and p-shapes.
combinatoricscombinatorial geometrytilesTiling
sum a/{1 + 2b^3} >= (a^2 + b^2 + c^2 + d^2}/3 if a,b,c,d>=0, a+b+c+d = 3

Source: 2017 Romania JBMO TST 2.4

6/27/2020
Let a,b,c,da, b, c, d be non-negative real numbers satisfying a+b+c+d=3a + b + c + d = 3. Prove that a1+2b3+b1+2c3+c1+2d3+d1+2a3≥a2+b2+c2+d23\frac{a}{1 + 2b^3} + \frac{b}{1 + 2c^3} +\frac{c}{1 + 2d^3} +\frac{d}{1 + 2a^3} \ge \frac{a^2 + b^2 + c^2 + d^2}{3} When does the equality hold?
inequalitiesalgebra
Combinatorics, equilateral triangles are divided

Source: JBMO TST 3 2017 P4

11/14/2018
The sides of an equilateral triangle are divided into n equal parts by n−1n-1 points on each side. Through these points one draws parallel lines to the sides of the triangle. Thus, the initial triangle is divides into n2n^2 equal equilateral triangles. In every vertex of such a triangle there is a beetle. The beetles start crawling simultaneously, with equal speed, along the sides of the small triangles. When they reach a vertex, the beetles change the direction of their movement by 60∘60^{\circ} or by 120∘120^{\circ} . a) Prove that, if n≥7n \geq 7, the beetles can move indefinitely on the sides of the small triangles without two beetles ever meeting in a vertex of a small triangle. b) Determine all the values of n≥1n \geq 1 for which the beetles can move along the sides of the small triangles without meeting in their vertices.
combinatorics
<NHP=<AHQ wanted, 2 equilateral triangles exterior to a right triangle

Source: 2017 Romania JBMO TST4 p4

5/16/2020
Let ABCABC be a right triangle, with the right angle at AA. The altitude from AA meets BCBC at HH and MM is the midpoint of the hypotenuse [BC][BC]. On the legs, in the exterior of the triangle, equilateral triangles BAPBAP and ACQACQ are constructed. If NN is the intersection point of the lines AMAM and PQPQ, prove that the angles ∠NHP\angle NHP and ∠AHQ\angle AHQ are equal.
Miguel Ochoa Sanchez and Leonard Giugiuc
Equilateralgeometryanglesequal angles
min no of red unit squares to be painting into a m x n board

Source: 2017 Romania JBMO TST 5.4

6/27/2020
Consider an m\times n board where m,n≥3m, n \ge 3 are positive integers, divided into unit squares. Initially all the squares are white. What is the minimum number of squares that need to be painted red such that each 3\times 3 square contains at least two red squares?
Andrei Eckstein and Alexandru Mihalcu
combinatorial geometrycombinatoricsColoring