MathDB

[url=https://artofproblemsolving.com/community/c677808]Cyprus IMO TST 2018
[url=https://artofproblemsolving.com/community/c6h1666662p10591751]Problem 1. Determine all integers

n \geq 2

for which the number

11111

in base

n

is a perfect square.
[url=https://artofproblemsolving.com/community/c6h1666663p10591753]Problem 2. Consider a trapezium

AB \Gamma \Delta

, where

A\Delta \parallel B\Gamma

and

\measuredangle A = 120^{\circ}

. Let

E

be the midpoint of

AB

and let

O_1

and

O_2

be the circumcenters of triangles

AE \Delta

and

BE\Gamma

, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle

O_1 E O_2

.
[url=https://artofproblemsolving.com/community/c6h1666660p10591747]Problem 3. Find all triples

(\alpha, \beta, \gamma)

of positive real numbers for which the expression

K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}

obtains its minimum value.
[url=https://artofproblemsolving.com/community/c6h1666661p10591749]Problem 4. Let

\Lambda= \{1, 2, \ldots, 2v-1,2v\}

and

P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}

be a permutation of the elements of

\Lambda

.
(a) Prove that

\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.

(b) Determine the largest positive integer

m

such that we can partition the

m\times m

square into

7

rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence:

1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Source

Problems(1)