2016 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

(8)

2

Easy sequence

(a_n)

a_1=0

a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.

a_{2016}>{1\over 2}+a_{1000}

.

tuymaada 2016/junior P1

Tanya and Serezha have a heap of

2016

candies. They make moves in turn, Tanya moves first. At each move a player can eat either one candy or (if the number of candies is even at the moment) exactly half of all candies. The player that cannot move loses. Which of the players has a winning strategy?

2

Graph problem

A connected graph is given. Prove that its vertices can be coloured blue and green and some of its edges marked so that every two vertices are connected by a path of marked edges, every marked edge connects two vertices of different colour and no two green vertices are connected by an edge of the original graph.

Nice problem

The flights map of air company

K_{r,r}

presents several cities. Some cities are connected by a direct (two way) flight, the total number of flights is m. One must choose two non-intersecting groups of r cities each so that every city of the first group is connected by a flight with every city of the second group. Prove that number of possible choices does not exceed

2*m^r

2

Two prime numbers

The ratio of prime numbers

p

q

does not exceed 2 (

p\ne q

). Prove that there are two consecutive positive integers such that the largest prime divisor of one of them is

p

and that of the other is

q

A problem

Positive numbers are written in the squares of a 10 × 10 table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to

10^2

. Then the frogs jumped again, and the sum changed to

10^3

and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?

2

Covering

A cube stands on one of the squares of an infinite chessboard. On each face of the cube there is an arrow pointing in one of the four directions parallel to the sides of the face. Anton looks on the cube from above and rolls it over an edge in the direction pointed by the arrow on the top face. Prove that the cube cannot cover any

5\times 5

Easy geo problem

D

on the altitude

AA_1

of an acute triangle

ABC

\angle BDC=90^\circ

H

is the orthocentre of

ABC

. A circle with diameter

AH

is constructed. Prove that the tangent drawn from

B

to this circle is equal to

BD

1

Concurrent

AA_1

BB_1

CC_1

of an acute triangle

ABC

H

A_0

B_0

C_0

are the midpoints of

BC

CA

AB

respectively. Points

A_2

B_2

C_2

on the segments

AH

BH

HC_1

respectively are such that

\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ

. Prove that the lines

AC_2

BA_2

CB_2

are concurrent.

2

Russian geometry problem

a

b

c

d

0<a \leq b \leq d \leq c

{a+c=b+d}

. Prove that for every internal point

P

of a segment with length

a

this segment is a side of a circumscribed quadrilateral with consecutive sides

a

b

c

d

, such that its incircle contains~

P

existance of odd digital number

Is there a positive integer

N>10^{20}

such that all its decimal digits are odd, the numbers of digits 1, 3, 5, 7, 9 in its decimal representation are equal, and it is divisible by each 20-digit number obtained from it by deleting digits? (Neither deleted nor remaining digits must be consecutive.)

2

Equation dio

For each positive integer

k

find the number of solutions in nonnegative integers

x,y,z

x\le y \le z

of the equation

8^k=x^3+y^3+z^3-3xyz

Three variable inequality for juniors

Non-negative numbers

a

b

c

a^2+b^2+c^2\geq 3

. Prove the inequality

(a+b+c)^3\geq 9(ab+bc+ca).

1

Nonsymmetric Inequality

x

y

z>{3\over 2}

prove the inequality x^{24} + \root 5\of {y^{60}+z^{40}} \geq \left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2.