MathDB

2018 All-Russian Olympiad

Part of All-Russian Olympiad

Subcontests

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Encoding order of cards

In a card game, each card is associated with a numerical value from 1 to 100, with each card beating less, with one exception: 1 beats 100. The player knows that 100 cards with different values lie in front of him. The dealer who knows the order of these cards can tell the player which card beats the other for any pair of cards he draws. Prove that the dealer can make one hundred such messages, so that after that the player can accurately determine the value of each card.

All Russian MO 2018 Grade 11 P7

Given a sequence of positive integers a1,a2,a3,...a_1,a_2,a_3,... defined by an=⌊n20182017⌋a_n=\lfloor n^{\frac{2018}{2017}}\rfloor. Show that there exists a positive integer NN such that among any NN consecutive terms in the sequence, there exists a term whose decimal representation contain digit 55.
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Tangent circles and isosceles triangle

Circle ω\omega is tangent to sides AB,ACAB, AC of triangle ABCABC. A circle Ω\Omega touches the side ACAC and line ABAB (produced beyond BB), and touches ω\omega at a point LL on side BCBC. Line ALAL meets ω,Ω\omega, \Omega again at K,MK, M. It turned out that KB∥CMKB \parallel CM. Prove that △LCM\triangle LCM is isosceles.

All Russian Olympiad 2018 Day1 P2

Let n≥2n\geq 2 and x1,x2,…,xnx_{1},x_{2},\ldots,x_{n} positive real numbers. Prove that 1+x121+x1x2+1+x221+x2x3+⋯+1+xn21+xnx1≥n.\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.
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All-Russian Olympiad Day 2 Problem 10.7.

ABCDABCD is a convex quadrilateral. Angles AA and CC are equal. Points MM and NN are on the sides ABAB and BCBC such that MN∣∣ADMN||AD and MN=2ADMN=2AD. Let KK be the midpoint of MNMN and HH be the orthocenter of △ABC\triangle ABC. Prove that HKHK is perpendicular to CDCD.

All-Russian MO 2018 Grade 10 P8

The board used for playing a game consists of the left and right parts. In each part there are several fields and there’re several segments connecting two fields from different parts (all the fields are connected.) Initially, there is a violet counter on a field in the left part, and a purple counter on a field in the right part. Lyosha and Pasha alternatively play their turn, starting from Pasha, by moving their chip (Lyosha-violet, and Pasha-purple) over a segment to other field that has no chip. It’s prohibited to repeat a position twice, i.e. can’t move to position that already been occupied by some earlier turns in the game. A player losses if he can’t make a move. Is there a board and an initial positions of counters that Pasha has a winning strategy?

All-Russian MO 2018 Grade 11 P8

Initially, on the lower left and right corner of a 2018×20182018\times 2018 board, there're two horses, red and blue, respectively. AA and BB alternatively play their turn, AA start first. Each turn consist of moving their horse (AA-red, and BB-blue) by, simultaneously, 2020 cells respect to one coordinate, and 1717 cells respect to the other; while preserving the rule that the horse can't occupied the cell that ever occupied by any horses in the game. The player who can't make the move loss, who has the winning strategy?
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Paintful game on a circle

On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?

All-Russian Olympiad Day 2 Problem 10.5.

In a 10×1010\times 10 table, positive numbers are written. It is known that, looking left-right, the numbers in each row form an arithmetic progression and, looking up-down, the numbers is each column form a geometric progression. Prove that all the ratios of the geometric progressions are equal.

All-Russian MO 2018 Grade 11 P5

On the table, there're 10001000 cards arranged on a circle. On each card, a positive integer was written so that all 10001000 numbers are distinct. First, Vasya selects one of the card, remove it from the circle, and do the following operation: If on the last card taken out was written positive integer kk, count the kthk^{th} clockwise card not removed, from that position, then remove it and repeat the operation. This continues until only one card left on the table. Is it possible that, initially, there's a card AA such that, no matter what other card Vasya selects as first card, the one that left is always card AA?
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Knight tour

On the n×nn\times n checker board, several cells were marked in such a way that lower left (LL) and upper right(RR) cells are not marked and that for any knight-tour from LL to RR, there is at least one marked cell. For which n>3n>3, is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?

All Russian Olympiad Day 1 P4

On the sides ABAB and ACAC of the triangle ABCABC, the points PP and QQ are chosen, respectively, so that PQ∥BCPQ\parallel BC. Segments BQBQ and CPCP intersect at point OO. Point A′A' is symmetric to point AA relative to line BCBC. The segment A′OA'O intersects the circumcircle ww of the triangle APQAPQ at the point SS. Prove that circumcircle of BSCBSC is tangent to the circle ww.

All-Russian Olympiad Day 1 Problem 10.4.

Initially, a positive integer is written on the blackboard. Every second, one adds to the number on the board the product of all its nonzero digits, writes down the results on the board, and erases the previous number. Prove that there exists a positive integer which will be added inifinitely many times.
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All Russian Olympiad 2018 Day1 P3

Suppose that a1,⋯ ,a25 a_1,\cdots , a_{25} are non-negative integers, and k k is the smallest of them. Prove that [a1]+[a2]+⋯+[a25]≥[a1+a2+⋯+a25+200k].\big[\sqrt{a_1}\big]+\big[\sqrt{a_2}\big]+\cdots+\big[\sqrt{a_{25}}\big ]\geq\big[\sqrt{a_1+a_2+\cdots+a_{25}+200k}\big]. (As usual, [x][x] denotes the integer part of the number xx , that is, the largest integer not exceeding xx.)

All-Russian Olympiad Day 1 Problem 10.3.

A positive integer kk is given. Initially, NN cells are marked on an infinite checkered plane. We say that the cross of a cell AA is the set of all cells lying in the same row or in the same column as AA. By a turn, it is allowed to mark an unmarked cell AA if the cross of AA contains at least kk marked cells. It appears that every cell can be marked in a sequence of such turns. Determine the smallest possible value of NN.
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Sequence is all prime

Suppose a1,a2,…a_1,a_2, \dots is an infinite strictly increasing sequence of positive integers and p1,p2,…p_1, p_2, \dots is a sequence of distinct primes such that pn∣anp_n \mid a_n for all n≥1n \ge 1. It turned out that an−ak=pn−pka_n-a_k=p_n-p_k for all n,k≥1n,k \ge 1. Prove that the sequence (an)n(a_n)_n consists only of prime numbers.

All Russian Olympiad Day 1 P1

The polynomial P(x)P (x) is such that the polynomials P(P(x))P (P (x)) and P(P(P(x)))P (P (P (x))) are strictly monotone on the whole real axis. Prove that P(x)P (x) is also strictly monotone on the whole real axis.

All-Russian Olympiad Day 1 Problem 10.1.

Determine the number of real roots of the equation ∣x∣+∣x+1∣+⋯+∣x+2018∣=x2+2018x−2019|x|+|x+1|+\cdots+|x+2018|=x^2+2018x-2019