MathDB

3

Part of 2015 Iran Team Selection Test

Problems(3)

Rectangles into each other

Source: Iran TST 2015 , Test 2 day 1 P3

8/18/2023
Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes such that each two have an area equal to 1 in common.
combinatorics
Sequence of numbers in form of a^2+b^2

Source: Iran TST 2015, exam 1, day 1 problem 3

5/11/2015
Let b1<b2<b3<… b_1<b_2<b_3<\dots be the sequence of all natural numbers which are sum of squares of two natural numbers. Prove that there exists infinite natural numbers like mm which bm+1−bm=2015b_{m+1}-b_m=2015 .
number theory
iran TST

Source: iran TST 2015 third exam p3

6/6/2015
a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n are 2n2n positive real numbers such that a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n aren't all equal. And assume that we can divide a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n into two subsets with equal sums.similarly b1,b2,⋯ ,bnb_1,b_2,\cdots ,b_n have these two conditions. Prove that there exist a simple 2n2n-gon with sides a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n and parallel to coordinate axises Such that the lengths of horizontal sides are among a1,a2,⋯ ,ana_1,a_2,\cdots ,a_n and the lengths of vertical sides are among b1,b2,⋯ ,bnb_1,b_2,\cdots ,b_n.(simple polygon is a polygon such that it doesn't intersect itself)
combinatorics