The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than (a) $0.7$ km? (b) $0.8$ km? (Grigory Kondakov, Moscow) You may assume that the banks consist of segments and arcs of circles.

In a convex quadrilateral $ABCD$ suppose $AC \cap BD = O$ and $M$ is the midpoint of $BC$. Let $MO \cap AD = E$. Prove that $\frac{AE}{ED} = \frac{S_{\triangle ABO}}{S_{\triangle CDO}}$.

Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$

[b]a)[/b] Show that, for any distinct natural numbers $ m,n, $ the rings $ \mathbb{Z}_2\times \underbrace{\cdots}_{m\text{ times}} \times\mathbb{Z}_2,\mathbb{Z}_2\times \underbrace{\cdots}_{n\text{ times}} \times\mathbb{Z}_2 $ are homomorphic, but not isomorphic. [b]b)[/b] Show that there are infinitely many pairwise nonhomomorphic rings of same order.

Inside the triangle $A_1A_2A_3$ with sides $a_1$, $a_2$, $a_3$, three points are given, which we label $P_1$, $P_2$, $P_3$ so that the product of their distances from the corresponding sides $a_1$, $a_2$, $a_3$ is as large as possible. Prove that the triangles $P_1A_2A_3$, $A_1P_2A_3$, $A_1A_2P_3$ cover the triangle. [hide=original wording]V trojúhelníku A1A2A3 se stranami a1, a2, a3 jsou dány tři body, které označíme Pi, P2, P3 tak, aby součin jejich vzdáleností od odpovídajících stran a1, a2, a3 byl co největší. Dokažte, že trojúhelníky P1A2A3, A1P2A3, A1A2P3 pokrývají trojúhelník.[/quote]

A $7 \times 7$ square is divided into unit squares by lines parallel to its sides. Some Swiss crosses (obtained by removing corner unit squares from a square of side $3$) are to be put on the large square, with the edges along division lines. Find the smallest number of unit squares that need to be marked in such a way that every cross covers at least one marked square.

Let $n\ge5$ be an integer. A trapezoid with base lengths of $1$ and $r$ is tiled by $n$ (not necessarily congruent) equilateral triangles. In terms of $n$, find the maximum possible value of $r$. [i]Linus Tang[/i]

Given a positive integer $k\geq2$, set $a_1=1$ and, for every integer $n\geq 2$, let $a_n$ be the smallest solution of equation \[x=1+\sum_{i=1}^{n-1}\left\lfloor\sqrt[k]{\frac{x}{a_i}}\right\rfloor\] that exceeds $a_{n-1}$. Prove that all primes are among the terms of the sequence $a_1,a_2,\ldots$

Determine all pairs $(a,b)$ of positive integers satisfying \[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]

Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$, $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ($c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$). Prove: $\left| S - 10^k AB \right| \leq 9k.$

The product of $10$ natural numbers is equal to $10^{10}$. What is the largest possible sum of these numbers?

Is there a power of $2$ such that it is possible to rearrange the digits giving another power of $2$?

Given three pairwise tangent equal circles $\Omega_i (i=1,2,3)$ with radius $r$. The circle $\Gamma $ touches the three circles internally (circumscribed about 3 circles).The three equal circles $\omega_i (i=1,2,3)$ with radius $x$ touches $\Omega_i$ and $\Omega_{i+1}$ externally ($\Omega_4= \Omega_1$) and touches $\Gamma$ internally.Find $x$ in terms of $r$

Show that if the sides $a, b, c$ of a triangle satisfy the equation \[2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,\] then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.

Let $H_n=\sum_{r=1}^{n} \frac{1}{r}$. Show that $$n-(n-1)n^{-1\slash (n-1)}>H_n>n(n+1)^{1\slash n}-n$$ for $n>2$.

Let be a sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ chosen such that the limit of the sequence $ \left( x_{n+2011}-x_n \right)_{n\ge 1} $ exists. Calculate $ \lim_{n\to\infty } \frac{x_n}{n} . $ [i]Cosmin Nițu[/i]

Polly can do the following operations on a quadratic trinomial: 1) Swapping the places of its leading coefficient and constant coefficient (swapping $a_2$ with $a_0$); 2) Substituting (changing) $x$ with $x-m$, where $m$ is an arbitrary real number; Is it possible for Polly to get $25x^2+5x+2014$ from $6x^2+2x+1996$ with finite applications of the upper operations?

Find all positive integers $x$ for which there exists a positive integer $y$ such that $\dbinom{x}{y}=1999000$

[list] [*] There are $128$ coins of two different weights, $64$ each. How can one always find two coins of different weights by performing no more than $7$ weightings on a regular balance? [*] There are $8$ coins of two different weights, $4$ each. How can one always find two coins of different weights by performing two weightings on a regular balance?[/list]

Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$. Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.

Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.

In how many ways can the set $N ={1, 2, \cdots, n}$ be partitioned in the form $p(N) = A_{1}\cup A_{2}\cup \cdots \cup A_{k},$ where $A_{i}$ consists of elements forming arithmetic progressions, all with the same common positive difference $d_{p}$ and of length at least one? At least two? [hide="Solution"] [b]Part 1[/b] Claim: There are $2^{n}-2$ ways of performing the desired partitioning. Let $d(k)$ equal the number of ways $N$ can be partitioned as above with common difference $k.$ We are thus trying to evaluate $\sum_{i=1}^{n-1}d(i)$ [b]Lemma: $d(i) = 2^{n-i}$[/b] We may divide $N$ into various rows where the first term of each row denotes a residue $\bmod{i}.$ The only exception is the last row, as no row starts with $0$; the last row will start with $i.$ We display the rows as such: $1, 1+i, 1+2i, 1+3i, \cdots$ $2, 2+i, 2+2i, 2+3i, \cdots$ $\cdots$ $i, 2i, 3i, \cdots$ Since all numbers have one lowest remainder $\bmod{i}$ and we have covered all possible remainders, all elements of $N$ occur exactly once in these rows. Now, we can take $k$ line segments and partition a given row above; all entries within two segments would belong to the same set. For example, we can have: $1| 1+i, 1+2i, 1+3i | 1+4i | 1+5i, 1+6i, 1+7i, 1+8i,$ which would result in the various subsets: ${1},{1+i, 1+2i, 1+3i},{1+4i},{1+5i, 1+6i, 1+7i, 1+8i}.$ For any given row with $k$ elements, we can have at most $k-1$ segments. Thus, we can arrange any number of segments where the number lies between $0$ and $k-1$, inclusive, in: $\binom{k-1}{0}+\binom{k-1}{1}+\cdots+\binom{k-1}{k-1}= 2^{k-1}$ ways. Applying the same principle to the other rows and multiplying, we see that the result always gives us $2^{n-i},$ as desired. We now proceed to the original proof. Since $d(i) = 2^{n-i}$ by the above lemma, we have: $\sum_{i=1}^{n-1}d(i) = \sum_{i=1}^{n-1}2^{n-i}= 2^{n}-2$ Thus, there are $2^{n}-2$ ways of partitioning the set as desired. [b]Part 2[/b] Everything is the same as above, except the lemma slightly changes to $d(i) = 2^{n-i}-i.$ Evaluating the earlier sum gives us: $\sum_{i=1}^{n-1}d(i) = \sum_{i=1}^{n-1}2^{n-i}-i = 2^{n}-\frac{n(n-1)}{2}-2$ [/hide]

Let $n$ be a positive integer. Show that if p is prime dividing $5^{4n}-5^{3n}+5^{2n}-5^{n}+1$, then $p\equiv 1 \;(\bmod\; 4)$.

Determine whether the equation $$x^4+y^3=z!+7$$ has an infinite number of solutions in positive integers.