MathDB

P1

Part of Russian TST 2015

Problems(7)

No worms were hurt in the making of this problem

Source: Russian TST 2015, Day 7 P1

4/20/2023
A worm is called an adult if its length is one meter. In one operation, it is possible to cut an adult worm into two (possibly unequal) parts, each of which immediately becomes a worm and begins to grow at a speed of one meter per hour and stops growing once it reaches one meter in length. What is the smallest amount of time in which it is possible to get nn{} adult worms starting with one adult worm? Note that it is possible to cut several adult worms at the same time.
combinatorics
Japan 2005 reboot

Source: Russian TST 2015, Day 8 P1 (Group NG), P2 (Groups A & B)

4/21/2023
Let P(x,y)P(x, y) and Q(x,y)Q(x, y) be polynomials in two variables with integer coefficients. The sequences of integers a0,a1,…a_0, a_1,\ldots and b0,b1,…b_0, b_1,\ldots satisfy a_{n+1}=P(a_n,b_n),  b_{n+1}=Q(a_n,b_n)for all n⩾0n\geqslant 0. Let mnm_n be the number of integer points of the coordinate plane, lying strictly inside the segment with endpoints (an,bn)(a_n,b_n) and (an+1,bn+1)(a_{n+1},b_{n+1}). Prove that the sequence m0,m1,…m_0,m_1,\ldots is non-decreasing.
algebrapolynomiallattice points
Inequality with high-order roots

Source: Russian TST 2015, Day 8 P1 (Groups A & B)

4/21/2023
Let n>4n>4 be a natural number. Prove that ∑k=2nkk−1k<n.\sum_{k=2}^n\sqrt[k]{\frac{k}{k-1}}<n.
algebrainequalities
Only squares staisfy these conditions

Source: Russian TST 2015, Day 10 P1 (Group NG), P2 (Groups A &amp; B)

4/21/2023
The points A′,B′,C′,D′A', B', C', D' are selected respectively on the sides AB,BC,CD,DAAB, BC, CD, DA of the cyclic quadrilateral ABCDABCD. It is known that AA′=BB′=CC′=DD′AA' = BB' = CC' = DD' and ∠AA′D′=∠BB′A′=∠CC′B′=∠DD′C′.\angle AA'D' =\angle BB'A' =\angle CC'B' =\angle DD'C'.Prove that ABCDABCD is a square.
geometrysquare
Easy divisibility NT

Source: Russian TST 2015, Day 9 P1 (Groups A &amp; B)

4/21/2023
Find all pairs of natural numbers (a,b)(a,b) satisfying the following conditions:
[*]b−1b-1 is divisible by a+1a+1 and [*]a2+a+2a^2+a+2 is divisible by bb.
number theoryDivisibility
L-trominoes on board

Source: Russian TST 2015, Day 10 P1 (Group NG), P2 (Groups A &amp; B)

4/21/2023
A 2015×20152015\times2015 chessboard is given, the cells of which are painted white and black alternatively so that the corner cells are black. There are nn{} [url=https://i.stack.imgur.com/V1kdh.png]L-trominoes placed on the board, no two of which overlap and which cover all of the black cells. Find the smallest possible value of nn{}.
combinatoricsboard
Cute NT with bounding

Source: Russian TST 2015, Day 10 P1 (Groups A &amp; B)

4/21/2023
Prove that there exist two natural numbers a,ba,b such that ∣a−m∣+∣b−n∣>1000|a-m|+|b-n|>1000 for any relatively prime natural numbers m,nm,n.
number theoryprime numbers