For an integer $n$, define $f(n)$ to be the greatest integer $k$ such that $2^k$ divides $\binom{n}{m}$ for some $0 \le m \le n$. Compute $f(1) + f(2) + \cdots + f(2048)$.

Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have \[ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3). \] Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$.

[b]7.1[/b] Given a convex $n$-gon all of whose angles are obtuse. Prove that the sum of the lengths of the diagonals in it is greater than the sum of the lengths of the sides. [b]7.2[/b] Find all integer values for $x$ and $y$ such that $x^4 + 4y^4$ is a prime number[b]. (typo corrected)[/b] [b]7.3.[/b] Given a triangle $ABC$. Parallelograms $ABKL$, $BCMN$ and $ACFG$ are constructed on the sides, Prove that the segments $KN$, $MF$ and $GL$ can form a triangle. [img]https://cdn.artofproblemsolving.com/attachments/a/f/7a0264b62754fafe4d559dea85c67c842011fc.png[/img] [b]7.4 / 6.2[/b] Prove that a $10 \times 10$ chessboard cannot be covered with $ 25$ figures like [img]https://cdn.artofproblemsolving.com/attachments/0/4/89aafe1194628332ec13ad1c713bb35cbefff7.png[/img]. [b]7.5[/b] Find the greatest number of different natural numbers, each of which is less than $50$, and every two of which are coprime. [b]7.6.[/b] Given a triangle $ABC$.$ D$ and $E$ are the midpoints of the sides $AB$ and $BC$. Point$ M$ lies on $AC$ , $ME > EC$. Prove that $MD < AD$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/1dd901e0121e5c75a4039d21b954beb43dc547.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983461_1964_leningrad_math_olympiad]here[/url].

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

Thirty two pairs of identical twins are lined up in an $8\times 8$ formation. Prove that it is possible to choose $32 $ persons, one from each pair of twins, so that there is at least one chosen person in each row and in each column

Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).\] The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $

Find all polynomial with real coefficients such that: P(x^2+1)=P(x)^2+1

$ y$ varies inversely as the square of $ x$. When $ y\equal{}16$, $ x\equal{}1$. When $ x\equal{}8$, $ y$ equals: $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 128 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ 1024$

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$. [i]Proposed by Japan[/i]

There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k\plus{}1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k\plus{}2th$ move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$. Let $M$ be the midpoint of $EF$. Prove that $\angle AMC = 90^{\circ}$.

[b]2.[/b] Show that a ring $R$ has a unit element if and only if any $R$-module $G$ can be written as a direct sum of $RG$ and of the trivial submodule of $G$. (An $R$-module is a linear space with $R$ as its scalar domain. $RG$ denotes the submodule generated by the elements of the form $rg$($r \in R, g \in G$). The trivial submodule of $G$ consists of the elements $g$ of $G$ for which $rg=0$ holds for every $r \in R$.) [b](A. 20)[/b]

Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?

Determine all integers $n$ for which there exist an integer $k\geq 2$ and positive integers $x_1,x_2,\hdots,x_k$ so that $$x_1x_2+x_2x_3+\hdots+x_{k-1}x_k=n\text{ and } x_1+x_2+\hdots+x_k=2019.$$

Solve the system of equations: $ \sqrt {3x}(1 \plus{} \frac {1}{x \plus{} y}) \equal{} 2$ $ \sqrt {7y}(1 \minus{} \frac {1}{x \plus{} y}) \equal{} 4\sqrt {2}$

Given are the points $A$ and $B$ on the edge of a circular pool. The athlete has to get from point $A$ to point $B$ by walking along the edge of the pool or swimming in the pool; he can change the way he moves many times. How should an athlete move to get from point A to B in the shortest time, given that he moves twice as slowly in water as on land?

Let $n$ be a natural number divisible by $4$. Determine the number of bijections $f$ on the set $\{1,2,...,n\}$ such that $f (j )+f^{-1}(j ) = n+1$ for $j = 1,..., n.$

Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$. [i]Proposed by Justin Hsieh[/i]

Determine the maximal size of a set of positive integers with the following properties: $1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$. $2.$ No digit occurs more than once in the same integer. $3.$ The digits in each integer are in increasing order. $4.$ Any two integers have at least one digit in common (possibly at different positions). $5.$ There is no digit which appears in all the integers.

Exactly $2^{1012}$ of the subsets of $\{1, 2, \ldots, 2023\}$ are colored red. Is it always true that there exist three distinct red sets $A$, $B$, and $C$ such that every element of $A$ belongs to at least one of $B$ or $C$?

Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions $i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct). $ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$

The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $1, 1\slash 2, 1\slash 3, \ldots.$ Each segment makes with the preceding a given angle $\theta.$ What is the distance and what is the direction of the limiting points (if there is one) from the initial point of the first segment?

Let $S$ be the set of vertices of a regular $24$-gon. Find the number of ways to draw $12$ segments of equal lengths so that each vertex in $S$ is an endpoint of exactly one of the $12$ segments.

Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]