2018 Moscow Mathematical Olympiad

Grade 11

1

The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this square trinomial has?

2

There is tetrahedron and square pyramid, both with all edges equal $1$. Show how to cut them into several parts and glue together from these parts a cube (without voids and cracks, all parts must be used)

3

Are there such natural $n$, that exist polynomial of degree $n$ and with $n$ different real roots, and a) $P(x)P(x+1)=P(x^2)$ b) $P(x)P(x+1)=P(x^2+1)$

4

Are there natural solution of $$a^3+b^3=11^{2018}$$?

5

On the sides of the convex hexagon $ABCDEF$ into the outer side were built equilateral triangles $ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$. The triangle $B_1D_1F_1$ is equilateral too. Prove that, the triangle $A_1C_1E_1$ is also equilateral.

6

There is house with $2^n$ rooms and every room has one light bulb and light switch. But wiring was connected wrong, so light switch can turn on light in some another room. Master want to find what switch connected to every light bulb. He use next practice: he send some workers in the some rooms, then they turn on switches in same time, then they go to master and tell him, in what rooms light bulb was turned on. a) Prove that $2n$ moves is enough to find, how switches are connected to bulbs. b) Is $2n-1$ moves always enough ?

7

$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$

8

$2018\times 2018$ field is covered with $1 \times 2$ dominos, such that every $2 \times 2$ or $1 \times 4,4 \times 1$ figure is not covered by only two dominos. Can be covered more than $99\%$ of field ?

9

$x$ and $y$ are integer $5$-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also $\tan{x}-\tan{y}=1+\tan{x}\tan{y}$, where $x,y$ are angles in degrees. Find maximum of $x$

10

$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$

11

Ivan want to paint ball. Ivan can put ball in the glass with some paint, and then one half of ball will be painted. Ivan use $5$ glasses to paint glass competely. Prove, that one glass was not needed, and Ivan can paint ball with $4$ glasses, putting ball in it by same way.

Grade 10

1

Is there a number in the decimal notation of the square which has a sequence of digits "$2018$"?

2

In there $2018\times 2018$ square cells colored in white or black. It is known, that exists $10 \times 10$ square with only white cells and $10\times 10$ square with only black cells. For what minimal $d$ always exists square $10\times 10$ such that the number of black and white cells differs by no more than $d$?

3

$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.

4

We call the arrangement of $n$ ones and $m$ zeros around the circle as good, if we can swap neighboring zero and one in such a way that we get an arrangement, that differs from the original by rotation. For what natural $m$ and $n$ does a good arrangement exist?

5

We have a blue triangle. In every move, we divide the blue triangle by angle bisector to $2$ triangles and color one triangle in red. Prove, that after some moves we color more than half of the original triangle in red.

6

We divide $999\times 999$ square into the angles with $3$ cells. Prove, that number of ways is divided by $2^7$.( Angle is a figure, that we can get if we remove one cell from $2 \times 2$ square).

Grade 9

1

$a_1,a_2,...,a_{81}$ are nonzero, $a_i+a_{i+1}>0$ for $i=1,...,80$ and $a_1+a_2+...+a_{81}<0$. What is sign of $a_1*a_2*...*a_{81}$?

2

We have $4$ sticks. It is known, that for every $3$ sticks we can build a triangle with the same area. Is it true, that sticks have the same length?

3

$a_1,a_2,...,a_k$ are positive integers and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1$. Prove that equation $$[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n$$has no more than $a_1*a_2*...*a_k$ postivie integer solutions in $n$.

4

$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.

6

There are $2018$ peoples. We call the group of people as "club" if all members of same "club" are all friends, but not friends with a nonmember of "club". Prove, that we can divide peoples for $90$ rooms, such that no one room has all members of some "club".