Arranged in a circle are $2010$ digits, each of them equal to $1$, $2$, or $3$. For each positive integer $k$, it's known that in any block of $3k$ consecutive digits, each of the digits appears at most $k+10$ times. Prove that there is a block of several consecutive digits with the same number of $1$s, $2$s, and $3$s.

Let $n$ be a positive integer. Humanity will begin to colonize Mars. The SpaceY and SpaceZ agencies will be responsible for traveling between the planets. To prevent the rockets from colliding, they will travel alternately, with SpaceY making the first trip. On each trip, the responsible agency will do one of two types of mission: (i) choose a positive integer $k$ and take $k$ people to Mars, creating a new colony on the planet and settling them in that colony; (ii) choose some existing colony on Mars and a positive integer $k$ strictly smaller than the population of that colony, and bring $k$ people from that colony back to Earth. To maintain the organization on Mars, a mission cannot result in two colonies with the same population and the number of colonies must be at most $n$. The first agency that cannot carry out a mission will go bankrupt. Determine, in terms of $n$, which agency can guarantee that it will not go bankrupt first.

Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r>1$, are of the form $n=2^l~$ for some $l\geqslant 0$.

In the film there is $n$ roles. For each $i$ ($1 \le i \le n$), the role of number $i$ can play $a_i$ a person, and one person can play only one role. Every day a casting is held, in which participate people for $n$ roles, and from each role only one person. Let $p$ be a prime number such that $p \ge a_1, \ldots, a_n, n$. Prove that you can have $p^k$ castings such that if we take any $k$ people who are tried in different roles, they together participated in some casting ($k$ is a natural number not exceeding $n$ ).

Color every vertex of $2008$-gon with two colors, such that adjacent vertices have different color. If sum of angles of vertices of first color is same as sum of angles of vertices of second color, than we call $2008$-gon as interesting. Convex $2009$-gon one vertex is marked. It is known, that if remove any unmarked vertex, then we get interesting $2008$-gon. Prove, that if we remove marked vertex, then we get interesting $2008$-gon too.

Evaluate $ \int_{\frac{\pi}{8}}^{\frac{3}{8}\pi} \frac{11\plus{}4\cos 2x \plus{}\cos 4x}{1\minus{}\cos 4x}\ dx.$

Prove the inequality \[\sum_{i<j} \frac{a_ia_j}{a_i \plus{} a_j} \le \frac{n}{2(a_1 \plus{} a_2 \plus{}\cdots \plus{} a_n)}\sum_{i<j} a_ia_j\] for all positive real numbers $ a_1, a_2,\ldots , a_n$.

A loader has two carts. One of them can carry up to 8 kg, and another can carry up to 9 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 17 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry on his two carts, regardless of particular weights of sacks? [i]Author: M.Ivanov, D.Rostovsky, V.Frank[/i]

A cube consisting of $(2N)^3$ unit cubes is pierced by several needles parallel to the edges of the cube (each needle pierces exactly $2N$ unit cubes). Each unit cube is pierced by at least one needle. Let us call any subset of these needles “regular” if there are no two needles in this subset that pierce the same unit cube. a) Prove that there exists a regular subset consisting of $2N^2$ needles such that all of them have either the same direction or two different directions. b) What is the maximum size of a regular subset that does exist for sure? (Nikita Gladkov, Alexandr Zimin)

Let $d(n)$ be the number of positive integers that divide the integer $n$. For all positive integral divisors $k$ of $64800$, what is the sum of numbers $d(k)$? $ \textbf{(A)}\ 1440 \qquad\textbf{(B)}\ 1650 \qquad\textbf{(C)}\ 1890 \qquad\textbf{(D)}\ 2010 \qquad\textbf{(E)}\ \text{None of above} $

Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\rightarrow\mathbb{N}$ by $T(2k)=k$ and $T(2k+1)=2k+2$. We write $T^2(n)=T(T(n))$ and in general $T^k(n)=T^{k-1}(T(n))$ for any $k>1$. (i) Show that for each $n\in\mathbb{N}$, there exists $k$ such that $T^k(n)=1$. (ii) For $k\in\mathbb{N}$, let $c_k$ denote the number of elements in the set $\{n: T^k(n)=1\}$. Prove that $c_{k+2}=c_{k+1}+c_k$, for $k\ge 1$.

Consider the statements: $ \textbf{(1)}\ \text{p and q are both true} \qquad \textbf{(2)}\ \text{p is true and q is false} \qquad \textbf{(3)}\ \text{p is false and q is true} \qquad \textbf{(4)}\ \text{p is false and q is false.}$ How many of these imply the negative of the statement "p and q are both true?" $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Triangle $A_1B_1C_1$ is an equilateral triangle with sidelength $1$. For each $n>1$, we construct triangle $A_nB_nC_n$ from $A_{n-1}B_{n-1}C_{n-1}$ according to the following rule: $A_n,B_n,C_n$ are points on segments $A_{n-1}B_{n-1},B_{n-1}C_{n-1},C_{n-1}A_{n-1}$ respectively, and satisfy the following: \[\dfrac{A_{n-1}A_n}{A_nB_{n-1}}=\dfrac{B_{n-1}B_n}{B_nC_{n-1}}=\dfrac{C_{n-1}C_n}{C_nA_{n-1}}=\dfrac1{n-1}\] So for example, $A_2B_2C_2$ is formed by taking the midpoints of the sides of $A_1B_1C_1$. Now, we can write $\tfrac{|A_5B_5C_5|}{|A_1B_1C_1|}=\tfrac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n$. (For a triangle $\triangle ABC$, $|ABC|$ denotes its area.)

Prove that for coprime each positive integers $a, c$ there is a positive integer $b$ such that $c$ divides $\underbrace{b^{b^{b^{\ldots^b}}}}_\text{b times}-a$

Let $S$ be the set of points $A$ in the Cartesian plane such that the four points $A$, $(2, 3)$, $(-1, 0)$, and $(0, 6)$ form the vertices of a parallelogram. Let $P$ be the convex polygon whose vertices are the points in $S$. What is the area of $P$?

Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$ [i](8 points)[/i]

The intersection point $I$ of the angles bisectors of the triangle $ABC$ has reflections the points $P,Q,T$ wrt the triangle's sides . It turned out that the circle $s$ circumscribed around of the triangle $PQT$ , passes through the vertex $A$. Find the radius of the circumscribed circle of triangle $ABC$ if $BC = a$. (Gryhoriy Filippovskyi)

Suppose that $X$ and $Y$ are two metric spaces and $f:X \longrightarrow Y$ is a continious function. Also for every compact set $K \subseteq Y$, it's pre-image $f^{pre}(K)$ is a compact set in $X$. Prove that $f$ is a closed function, i.e for every close set $C\subseteq X$, it's image $f(C)$ is a closed subset of $Y$.

Given an acute triangle $ABC$ with altitudes $AD$ and $BE$ intersecting at $H$, $M$ is the midpoint of $AB$. A nine-point circle of $ABC$ intersects with a circumcircle of $ABH$ on $P$ and $Q$ where $P$ lays on the same side of $A$ (with respect to $CH$). Prove that $ED, PH, MQ$ are concurrent on circumcircle $ABC$

On a natural number $n$ you are allowed to operations : $(1)$ multiply $n$ by $2$ or $(2)$ subtract $3$ from $n$. For example starting with $8$ you can reach $13$ as follows : $8 \longrightarrow 16 \longrightarrow 13$. You need two steps and you cannot do in less than two steps. Starting from $11$, what is the least number of steps required to reach $121$?

[b]6.1 [/b] The bindery had 92 sheets of white paper and $135$ sheets of colored paper. It took a sheet of white paper to bind each book. and a sheet of colored paper. After binding several books of white Paper turned out to be half as much as the colored one. How many books were bound? [b]6.2[/b] Prove that if you multiply all the integers from $1$ to $1965$, you get the number, the last whose non-zero digit is even. [b]6.3[/b] The front tires of a car wear out after $25,000$ kilometers, and the rear tires after $15,000$ kilometers of travel. When should you swap tires so that they wear out at the same time? [b]6.4[/b] A rectangle $19$ cm $\times 65$ cm is divided by straight lines parallel to its sides into squares with side 1 cm. How many parts will this rectangle be divided into if you also draw a diagonal in it? [b]6.5[/b] Find the dividend, divisor and quotient in the example: [center][img]https://cdn.artofproblemsolving.com/attachments/2/e/de053e7e11e712305a89d3b9e78ac0901dc775.png[/img] [/center] [b]6.6[/b] Odd numbers from $1$ to $49$ are written out in table form $$\,\,\,1\,\,\,\,\,\, 3\,\,\,\,\,\, 5\,\,\,\,\,\, 7\,\,\,\,\,\, 9$$ $$11\,\,\, 13\,\,\, 15\,\,\, 17\,\,\, 19$$ $$21\,\,\, 23\,\,\, 25\,\,\, 27\,\,\, 29$$ $$31\,\,\, 33\,\,\, 35\,\,\, 37\,\,\, 39$$ $$41\,\,\, 43\,\,\, 45\,\,\, 47\,\,\, 49$$ $5$ numbers are selected, any two of which are not on the same line or in one column. What is their sum? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988081_1965_leningrad_math_olympiad]here[/url].

Solve the equation: $$x \cdot 2^{\dfrac{1}{x}}+\dfrac{1}{x} \cdot 2^x=4$$

Let $ABC$ be an acute triangle and $T$ be a point interior to this triangle. that $\angle ATB = \angle BTC = \angle CTA$. Let $M,N$ and $P$ be the feet of the perpendiculars from $T$ to $BC$, $CA$ and $AB$ respectively. Prove that if the circle circumscribed around $\vartriangle MNP$ cuts again the sides $ BC$, $CA$ and $AB$ in $M_1$, $N_1$, $P_1$ respectively, then the $\vartriangle M_1N_1P_1$ It is equilateral.

Ebeneezer is painting the edges of a cube. He wants to paint the edges so that the colored edges form a loop that does not intersect itself. For example, the loop should not look like a “figure eight” shape. If two colorings are considered equivalent if there is a rotation of the cubes so that the colored edges are the same, what is the number of possible edge colorings?

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.