MathDB

P2

Part of 2019 India IMO Training Camp

Problems(5)

Numbers not power of 5

Source: Indian TST D1 P2

7/17/2019
Show that there do not exist natural numbers a1,a2,…,a2018a_1, a_2, \dots, a_{2018} such that the numbers (a1)2018+a2,(a2)2018+a3,…,(a2018)2018+a1 (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 are all powers of 55 Proposed by Tejaswi Navilarekallu
number theory
Putting digits in dominoes

Source: Indian TST D2 P2

7/17/2019
Let nn be a natural number. A tiling of a 2n×2n2n \times 2n board is a placing of 2n22n^2 dominos (of size 2×12 \times 1 or 1×21 \times 2) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two sepearate tilings of a 2n×2n2n \times 2n board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours.
Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that nn is divisible by 33 Proposed by Tejaswi Navilarekallu
combinatoricsTiling
AO and KI meet on $\Gamma$

Source: Indian TST 3 P2

7/17/2019
Let ABCABC be an acute-angled scalene triangle with circumcircle Γ\Gamma and circumcenter OO. Suppose AB<ACAB < AC. Let HH be the orthocenter and II be the incenter of triangle ABCABC. Let FF be the midpoint of the arc BCBC of the circumcircle of triangle BHCBHC, containing HH.
Let XX be a point on the arc ABAB of Γ\Gamma not containing CC, such that ∠AXH=∠AFH\angle AXH = \angle AFH. Let KK be the circumcenter of triangle XIAXIA. Prove that the lines AOAO and KIKI meet on Γ\Gamma. Proposed by Anant Mudgal
anant mudgal geogeometryHi
Not Combi

Source: Indian TST 2019 Practice Test 2 P2

7/17/2019
Determine all positive integers mm satisfying the condition that there exists a unique positive integer nn such that there exists a rectangle which can be decomposed into nn congruent squares and can also be decomposed into m+nm+n congruent squares.
geometryrectangle
Perimeter inequality

Source: Indian TST 2019 Practice Test 1 P2

7/17/2019
Let ABCABC be a triangle with ∠A=∠C=30∘.\angle A=\angle C=30^{\circ}. Points D,E,FD,E,F are chosen on the sides AB,BC,CAAB,BC,CA respectively so that ∠BFD=∠BFE=60∘.\angle BFD=\angle BFE=60^{\circ}. Let pp and p1p_1 be the perimeters of the triangles ABCABC and DEFDEF, respectively. Prove that p≤2p1.p\le 2p_1.
geometryperimeterinequalities