Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Saint Petersburg Mathematical Olympiad
Given three quadratic trinomials $f, g, h$ without roots. Their elder coefficients are the same, and all their coefficients for x are different. Prove that there is a number $c$ such that the equations $f (x) + cg (x) = 0$ and $f (x) + ch (x) = 0$ have a common root.
2016 Saint Petersburg Mathematical Olympiad
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook.  At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?
2016 Saint Petersburg Mathematical Olympiad
On the side $AB$ of the non-isosceles triangle $ABC$, let the points $P$ and $Q$ be so that $AC = AP$ and $BC = BQ$. The perpendicular bisector of the segment $PQ$ intersects the angle bisector of the $\angle C$ at the point $R$ (inside the triangle). Prove that $\angle ACB +  \angle PRQ = 180^o$.
2016 Saint Petersburg Mathematical Olympiad
Two different prime numbers $p$ and $q$ differ in less than $2$ times. Prove that exists two consecutive natural numbers, such that largest prime divisor of first number is $p$, and  largest prime divisor of second number is $q$.
2016 Saint Petersburg Mathematical Olympiad
Kostya and Sergey play a game on a white strip of length 2016 cells. Kostya (he plays first) in one move should paint black over two neighboring white cells. Sergey should paint either one white cell either three neighboring white cells. It is forbidden to make a move, after which a white cell is formed the doesn't having any white neighbors. Loses the one that can make no other move. However, if all cells are painted, then Kostya wins. Who will win if he plays the right game (has a winning strategy)?
2016 Saint Petersburg Mathematical Olympiad
Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that angle bisector of $\angle GDF$ passes though the midpoint of $I_1I_2 $.
2016 Saint Petersburg Mathematical Olympiad
A sequence of $N$ consecutive positive integers is called good  if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many good  sequences?
2015 Saint Petersburg Mathematical Olympiad
$x,y$ are real numbers such that $$x^2+y^2=1    ,  20x^3-15x=3$$Find the value of $|20y^3-15y|$.(K. Tyshchuk)
2015 Saint Petersburg Mathematical Olympiad
$a,b>1$ - are naturals, and $a^2+b,a+b^2$ are primes. Prove $(ab+1,a+b)=1$
2015 Saint Petersburg Mathematical Olympiad
There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$
2015 Saint Petersburg Mathematical Olympiad
$ABCD$ is convex quadrilateral. Circumcircle of $ABC$ intersect $AD$ and $DC$ at points $P$ and $Q$. Circumcircle of $ADC$ intersect $AB$ and $BC$ at points $S$ and $R$. Prove that if $PQRS$ is parallelogram then $ABCD$ is parallelogram
2015 Saint Petersburg Mathematical Olympiad
Square with side 100 was cut by 99 horizontal and 99 vertical lines into 10000 rectangles (not necessarily with integer sides). How many rectangles in this square with area not exceeding 1 at least can be?
2015 Saint Petersburg Mathematical Olympiad
In country there are some cities, some pairs of cities are connected with roads.From every city go out exactly $100$ roads.  We call $10$ roads, that go out from same city, as bunch. Prove, that we can split all roads in several bunches.
2015 Saint Petersburg Mathematical Olympiad
Let $BL$ be angle bisector of acute triangle $ABC$.Point $K$ choosen on $BL$ such that $\measuredangle AKC-\measuredangle ABC=90º$.point $S$ lies on the extention of $BL$ from $L$ such that $\measuredangle ASC=90º$.Point $T$ is diametrically opposite the point $K$  on the circumcircle of $\triangle AKC$.Prove that $ST$ passes through midpoint of arc $ABC$.(S. Berlov)

Click to reveal hidden text:trampoline: my 100th post :trampoline:
2015 Saint Petersburg Mathematical Olympiad
There is child camp with some rooms. Call room as $4-$room, if $4$ children live here.  Not less then half of all rooms are  $4-$rooms , other rooms are $3-$rooms. Not less than $2/3$ girls live in $3-$rooms. Prove that not less than $35\%$ of all children are boys.
2015 Saint Petersburg Mathematical Olympiad
The beaver is chess piece that move to $2$ cells by horizontal or vertical.  Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?
2015 Saint Petersburg Mathematical Olympiad
$ABCD$ - convex quadrilateral. Bisectors of angles $A$ and $D$ intersect in $K$,  Bisectors of angles $B$ and $C$ intersect in $L$. Prove $$2KL \geq |AB-BC+CD-DA|$$
2015 Saint Petersburg Mathematical Olympiad
A positive integer $n$ is called Olympic, if there exists a quadratic trinomial with integer coeffecients $f(x)$ satisfying $f(f(\sqrt{n}))=0$. Determine, with proof, the largest Olympic number not exceeding $2015$.



A. Khrabrov
2015 Saint Petersburg Mathematical Olympiad
$ABCDE$ is convex pentagon. $\angle BCA=\angle BEA = \frac{\angle BDA}{2}, \angle BDC =\angle EDA$.

Prove, that $\angle DEB=\angle DAC$
2015 Saint Petersburg Mathematical Olympiad
A sequence of integers is defined as follows: $a_1=1,a_2=2,a_3=3$ and for $n>3$, $$a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.$$Prove that every natural number occurs exactly once in this sequence.



M. Ivanov
2015 Saint Petersburg Mathematical Olympiad
There is convex $n-$gon. We color all its sides and also diagonals, that goes out from one vertex. So we have $2n-3$ colored segments. We write positive numbers on  colored segments. In one move we can take quadrilateral $ABCD$ such, that $AC$ and all sides are colored, then remove $AC$ and color $BD$ with number $\frac{xz+yt}{w}$, where $x,y,z,t,w$ - numbers on $AB,BC,CD,DA,AC$. After some moves we found that all colored segments are same that was at beginning. Prove, that they have same number that was at beginning.
2015 Saint Petersburg Mathematical Olympiad
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?



A. Khrabrov
2015 Saint Petersburg Mathematical Olympiad
$AB=CD,AD \parallel BC$ and $AD>BC$. $\Omega$ is circumcircle of $ABCD$. Point $E$ is on $\Omega$ such that $BE \perp AD$. Prove that $AE+BC>DE$
2015 Saint Petersburg Mathematical Olympiad
All cells of $2015 \times 2015$ table colored in one of $4$ colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all $4$ colors in less than $51\%$ of total number of ways.
2015 Saint Petersburg Mathematical Olympiad
Positive numbers $x, y, z$ satisfy the condition $$xy + yz + zx + 2xyz = 1.$$Prove that $4x + y + z \ge 2.$



A. Khrabrov
2015 Saint Petersburg Mathematical Olympiad
There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible.



(D. Karpov)
2014 Saint Petersburg Mathematical Olympiad
Let $f(x)$ is such function, that $f(x)=1$  for integer $x$ and $f(x)=0$ for non integer $x$.

Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)
2014 Saint Petersburg Mathematical Olympiad
There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads.

Prove, that he need not more than $199$ days to destroy all roads in country.
2014 Saint Petersburg Mathematical Olympiad
$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$
2014 Saint Petersburg Mathematical Olympiad
Points $B_1,C_1$  are on $AC$ and $AB$ and $B_1C_1 \parallel BC$. Circumcircle of $ABB_1$ intersect $CC_1$ at $L$. Circumcircle $CLB_1$ is tangent to $AL$.

Prove $AL \leq \frac{AC+AC_1}{2}$
2014 Saint Petersburg Mathematical Olympiad
$M$ is infinite set of natural numbers. If $a,b, a\neq b$ are in $M$, then $a^b+2$ or $a^b-2$ ( or both) are in $M$. Prove that there is composite number in $M$
2014 Saint Petersburg Mathematical Olympiad
In the $n \times n$ table in every cell there is one child. Every child looks in neigbour cell. So every child sees ear or back of the head of neighbour.  What is minimal number children, that see  ear ?
2014 Saint Petersburg Mathematical Olympiad
$I$ - incenter , $M$- midpoint of arc $BAC$ of circumcircle, $AL$ - angle bisector of triangle $ABC$. $MI$ intersect circumcircle in $K$. Circumcircle of $AKL$ intersect $BC$ at $L$ and $P$.

Prove that $\angle AIP=90$
2014 Saint Petersburg Mathematical Olympiad
$f(x)$ is square polynomial and $a \neq b$ such that $f(a)=b,f(b)=a$. Prove that there is not other pair $(c,d)$ that $f(c)=d,f(d)=c$
2014 Saint Petersburg Mathematical Olympiad
There are $40$ points on the two parallel lines. We divide it to pairs, such that line segments, that connects point in pair, do not intersect each other ( endpoint from one segment cannot lies on another segment). Prove, that number of ways to do it is less than $3^{39}$
2014 Saint Petersburg Mathematical Olympiad
$D$ is inner point of triangle $ABC$. $E$ is on $BD$ and $CE=BD$. $\angle ABD=\angle ECD=10,\angle BAD=40,\angle CED=60$ Prove, that $AB>AC$
2014 Saint Petersburg Mathematical Olympiad
$a_1\geq a_2\geq... \geq a_{100n}>0$

If we take from $(a_1,a_2,...,a_{100n})$  some $2n+1$ numbers $b_1\geq b_2 \geq ... \geq b_{2n+1}$ then $b_1+...+b_n > b_{n+1}+...b_{2n+1}$ Prove, that

$$(n+1)(a_1+...+a_n)>a_{n+1}+a_{n+2}+...+a_{100n}$$
2014 Saint Petersburg Mathematical Olympiad
Incircle $\omega$ of $ABC$ touch $AC$ at $B_1$. Point $E,F$ on the $\omega$ such that $\angle AEB_1=\angle B_1FC=90$. Tangents to $\omega$ at $E,F$ intersects in $D$, and $B$ and $D$ are on different sides for line $AC$. $M$- midpoint of $AC$.

Prove, that $AE,CF,DM$ intersects at one point.
2014 Saint Petersburg Mathematical Olympiad
Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of  even roads. Prove that king of city can place military bases in some cities such that  there are not roads between these cities, but for every city without base we can go from city with base by no more than $1$ road.



PSI think it should be one more condition, like there is cycle that connect all cities
2014 Saint Petersburg Mathematical Olympiad
All angles of $ABC$ are in $(30,90)$. Circumcenter of $ABC$ is $O$ and circumradius is $R$. Point $K$ is projection of $O$ to angle bisector of $\angle B$, point $M$ is midpoint $AC$. It is known, that $2KM=R$. Find $\angle B$
2014 Saint Petersburg Mathematical Olympiad
$100$ deputies formed $450$ commissions. Each two commissions has no more than  three common deputies, and every $5$ - no more than one. Prove that, that there are $4$ commissions that has exactly one common deputy each.
2014 Saint Petersburg Mathematical Olympiad
We call a natural number venerable if the sum of all its divisors, including $1$, but not including the number itself, is $1$ less than this number. Find all the venerable numbers, some exact degree of which is also venerable.
2014 Saint Petersburg Mathematical Olympiad
On a cellular plane with a cell side equal to $1$, arbitrarily $100 \times 100$ napkin is thrown. It covers some nodes (the node lying on the border of a napkin, is also considered covered). What is the smallest number of lines (going not necessarily along grid lines) you can certainly cover all these nodes?
2014 Saint Petersburg Mathematical Olympiad
Points $A,B$ are on circle $\omega$. Points $C$ and $D$ are moved on the arc $AB$, such that $CD$ has constant length. $I_1,I_2$ - incenters of $ABC$ and $ABD$.

Prove that line $I_1I_2$ is tangent to some fixed circle.
2014 Saint Petersburg Mathematical Olympiad
Natural $a,b,c$ are pairwise prime. There is infinite table with one integer number in every cell. Sum of numbers in every $a \times a$, every $b \times b$, every $c \times c$ squares  is even.

Is it true, that every number in table must be even?
2013 Saint Petersburg Mathematical Olympiad
Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $.





F. Petrov
2013 Saint Petersburg Mathematical Olympiad
At the faculty of mathematics study $40$ boys and $10$ girls.  Every girl acquaintance with all boys, who older than her, or tall(higher) than her. Prove that there exist two boys such that the sets of acquainted-girls of the boys are same.
2013 Saint Petersburg Mathematical Olympiad
Let $M$ and $N$ are midpoint of edges $AB$ and $CD$ of the tetrahedron $ABCD$, $AN=DM$ and $CM=BN$. Prove that $AC=BD$.





S. Berlov
2013 Saint Petersburg Mathematical Olympiad
Find all pairs $(p,q)$ of prime numbers, such that $2p-1$, $2q-1$, $2pq-1$ are perfect square.





F. Petrov,  A. Smirnov
2013 Saint Petersburg Mathematical Olympiad
Let $x_1$, ... , $x_{n+1} \in [0,1] $ and $x_1=x_{n+1} $. Prove that $$ \prod_{i=1}^{n} (1-x_ix_{i+1}+x_i^2)\ge 1. $$





A. Khrabrov,  F. Petrov
2013 Saint Petersburg Mathematical Olympiad
Let $(I_b)$, $(I_c)$ are excircles of a triangle $ABC$. Given a circle $ \omega $ passes through $A$ and externally tangents to the circles $(I_b)$ and $(I_c)$ such that it intersects with $BC$ at points $M$, $N$.

Prove that $ \angle BAM=\angle CAN $.





A. Smirnov
2013 Saint Petersburg Mathematical Olympiad
In the language of wolves has two letters $F$ and $P$, any finite sequence which forms a word.   А word $Y$ is called 'subpart' of word $X$ if Y is obtained from X by deleting some letters (for example,   the word  $FFPF$ has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF).  Determine $n$ such that  the $n$ is the greatest number of 'subpart's can have n-letter word language of wolves.





F. Petrov,  V. Volkov
2013 Saint Petersburg Mathematical Olympiad
Call number $A$ as interesting if $A$ is divided by every number that can be received from $A$ by crossing some last digits. Find maximum interesting number with different digits.
2013 Saint Petersburg Mathematical Olympiad
in a convex quadrilateral $ABCD$ , $M,N$ are midpoints of $BC,AD$ respectively. If $AM=BN$ and  $DM=CN$ then prove that $AC=BD$.







S. Berlov
2013 Saint Petersburg Mathematical Olympiad
On a circle there are some black and white points (there are at least $12$ points). Each point has $10$ neighbors ($5$ left and $5$ right neighboring points), $5$ being black and $5$ white. Prove that the number of points on the circle is divisible by $4$.
2013 Saint Petersburg Mathematical Olympiad
There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.
2013 Saint Petersburg Mathematical Olympiad
Given quadrilateral $ABCD$ with $AB=BC=CD$. Let $AC\cap BD=O$, $X,Y$ are symmetry points of $O$ respect to midpoints of $BC$, $AD$, and $Z$ is intersection point of lines, which perpendicular bisects of  $AC$, $BD$. Prove that $X,Y,Z$ are collinear.
2013 Saint Petersburg Mathematical Olympiad
There are $85$ soldiers with different heigth and age. Every day commander chooses random soldier  and send him and also all soldiers that are taller and older than this soldier, or all soldiers that are lower and  younger than this soldier to color grass.

Prove that after $10$ days we can find two soldiers, that color grass at same days.
2013 Saint Petersburg Mathematical Olympiad
Let $a_1,a_2$ - two naturals, and $1<b_1<a_1,1<b_2<a_2$ and $b_1|a_1,b_2|a_2$. Prove that $a_1b_1+a_2b_2-1$ is not divided by $a_1a_2$
2013 Saint Petersburg Mathematical Olympiad
if $a^2+b^2+c^2+d^2=1$ prove that $$ (1-a)(1-b)\ge cd.  $$





A. Khrabrov
2013 Saint Petersburg Mathematical Olympiad
$ABC$ is triangle. $l_1$- line passes through $A$ and parallel to $BC$, $l_2$ -  line passes through $C$ and parallel to $AB$. Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$. $XY=AC$. What  value can take $\angle A- \angle C$ ?
2013 Saint Petersburg Mathematical Olympiad
There are $100$ glasses, with $101,102,...,200$ cents.Two players play next game. In every move they can take some cents from one glass, but after move should be different number of cents in every glass. Who will win with right strategy?
2013 Saint Petersburg Mathematical Olympiad
Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$.

Click to reveal hidden text(It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )

M. Antipov
2012 Saint Petersburg Mathematical Olympiad
$a,b,c$ are reals, such that every pair of equations of $x^3-ax^2+b=0,x^3-bx^2+c=0,x^3-cx^2+a=0$ has common root.

Prove $a=b=c$
2012 Saint Petersburg Mathematical Olympiad
We have big multivolume encyclopaedia about dogs  on the shelf in alphabetical order, each volume in its specially selected place. Near each place there is an instruction that prescribes one of four actions: to rearrange

this volume is one or two places left or right. If you simultaneously run all instructions, volumes will be placed in the same places in another order. The cynologist Dima performs all the instructions every morning. Once he discovered,

that the volume of "Bichons" stands still, which was initially occupied by the volume of "Terriers". Prove ,

that after some time the volume of "Mudies" will stand on the original place of the volume

"Poodles".
2012 Saint Petersburg Mathematical Olympiad
At the base of the pyramid $SABCD$ lies a convex quadrilateral $ABCD$, such that $BC * AD = BD * AC$. Also $ \angle ADS =\angle BDS ,\angle ACS =\angle BCS$.

Prove that the plane $SAB$ is perpendicular to the plane of the base.
2012 Saint Petersburg Mathematical Olympiad
$x_1,...,x_n$ are reals and $x_1^2+...+x_n^2=1$

Prove, that exists such $y_1,...,y_n$ and $z_1,...,z_n$ such that $|y_1|+...+|y_n| \leq 1$; $max(|z_1|,...,|z_n|) \leq 1$ and $2x_i=y_i+z_i$ for every $i$
2012 Saint Petersburg Mathematical Olympiad
$n \geq k$ -two natural numbers. $S$ -such natural, that have not less than $n$ divisors. All divisors of $S$ are written

in descending order. What minimal number of divisors can have number from $k$-th place ?
2012 Saint Petersburg Mathematical Olympiad
On the coordinate plane in the first quarter  there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.
2012 Saint Petersburg Mathematical Olympiad
Some cities of Russia are connected with some cities of Ukraine with international airlines. The Interstate Council for the Promotion of Migration intends to introduce a one-way traffic on each airline so that, by taking off from the city, it could no longer be returned in this city (using other one-way airlines). Prove that the number of ways to do this is not divided  by $3$.
2012 Saint Petersburg Mathematical Olympiad
$\begin{cases} x^3-ax^2+b^3=0 \\x^3-bx^2+c^3=0 \\ x^3-cx^2+a^3=0 \end{cases}$

Prove that system hasn`t solutions if $a,b,c$ are different.
2012 Saint Petersburg Mathematical Olympiad
Points $C,D$ are on side $BE$ of triangle $ABE$, such that $BC=CD=DE$. Points $X,Y,Z,T$ are circumcenters of $ABE,ABC,ADE,ACD$. Prove, that $T$ - centroid of $XYZ$
2012 Saint Petersburg Mathematical Olympiad
$25$ students are on exams. Exam consists of some questions with $5$ variants of answer.  Every two students gave same answer for not more than $1$ question. Prove, that there are not more than $6$ questions in exam.
2012 Saint Petersburg Mathematical Olympiad
Some notzero reals numbers are placed around circle. For every two neighbour numbers $a,b$ it true, that $a+b$ and $\frac{1}{a}+\frac{1}{b}$ are integer. Prove that there are not more than $4$ different numbers.
2012 Saint Petersburg Mathematical Olympiad
$S$ is natural, and $S=d_{1}>d_2>...>d_{1000000}=1$ are all divisors of $S$. What minimal number of divisors can have $d_{250}$?
2012 Saint Petersburg Mathematical Olympiad
$ABCD$ is parallelogram. Line $l$ is perpendicular to $BC$ at $B$. Two circles passes through $D,C$, such that $l$ is tangent in points $P$ and $Q$. $M$ - midpoint $AB$.

Prove that $\angle DMP=\angle DMQ$
2012 Saint Petersburg Mathematical Olympiad
Find all integer $b$ such that  $[x^2]-2012x+b=0$ has odd number of roots.
2012 Saint Petersburg Mathematical Olympiad
Natural $a,b,c$ are $>100$ and $(a,b,c)=1$. $c|a+b,a|b+c$ Find minimal $b$
2012 Saint Petersburg Mathematical Olympiad
$ABCD$ is inscribed. Bisector of angle between diagonals intersect $AB$ anc $CD$ at $X$ and $Y$. $M,N$ are midpoints of $AD,BC$. $XM=YM$ Prove, that $XN=YN$.
2012 Saint Petersburg Mathematical Olympiad
In the $100 \times 100$ table in every cell there is natural number. All numbers in same row or column are different.

Can be that for every square sum of numbers, that are in angle cells, is square number ?
2012 Saint Petersburg Mathematical Olympiad
$ABC$ is triangle. Point $L$ is inside $ABC$ and lies on bisector of $\angle B$. $K$ is on $BL$. $\angle KAB=\angle LCB= \alpha$. Point $P$ inside triangle is such, that $AP=PC$ and $\angle APC=2\angle AKL$.

Prove that $\angle KPL=2\alpha$
2012 Saint Petersburg Mathematical Olympiad
We have $2012$ sticks with integer length, and sum of length is $n$. We need to have sticks with lengths $1,2,....,2012$. For it we can break some sticks ( for example from stick with length $6$ we can get $1$ and $4$).

For what minimal $n$ it is always possible?
2011 Saint Petersburg Mathematical Olympiad
$f(x),g(x)$ - two square trinomials and $a,b,c,d$ - some reals. $f(a)=2,f(b)=3,f(c)=7,f(d)=10$ and $g(a)=16,g(b)=15,g(c)=11$ Find $g(d)$
2011 Saint Petersburg Mathematical Olympiad
$ABC$-triangle with circumcenter $O$ and $\angle B=30$. $BO$ intersect $AC$ at $K$. $L$ - midpoint of arc $OC$ of circumcircle $KOC$, that does not contains $K$. Prove, that $A,B,L,K$ are concyclic.
2011 Saint Petersburg Mathematical Olympiad
Can we build parallelepiped $6 \times 7 \times 7$ from $1 \times 1 \times 2$ bricks, such that we have same amount bricks of one of $3$ directions ?
2011 Saint Petersburg Mathematical Olympiad
Call integer number $x$ as far from squares and cubes, if for every integer $k$ it is true : $|x-k^2|>10^6,|x-k^3|>10^6$.

Prove, that there are infinitely many far from squares and cubes degrees of $2$
2011 Saint Petersburg Mathematical Olympiad
Let $M(n)$ and $m(n)$ are maximal and minimal proper divisors of $n$

Natural number $n>1000$ is on the board. Every minute we replace our number with $n+M(n)-m(n)$. If we get  prime, then process is stopped.

Prove that after some moves we will get number, that is not divisible by $17$
2011 Saint Petersburg Mathematical Olympiad
$ABCD$ - convex quadrilateral. $M$ -midpoint $AC$ and $\angle MCB=\angle CMD =\angle MBA=\angle MBC-\angle MDC$.

Prove, that $AD=DC+AB$