Found problems: 85335
The integers $1, 2, \cdots, n^2$ are placed on the fields of an $n \times n$ chessboard $(n > 2)$ in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most $n + 1$. What is the total number of such placements?
Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that
\[\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )\]
prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that
\[\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )\]
What is the $100th$ number in the arithmetic sequence: $ 1,5,9,13,17,21,25,... $
$ \text{(A)}\ 397\qquad\text{(B)}\ 399\qquad\text{(C)}\ 401\qquad\text{(D)}\ 403\qquad\text{(E)}\ 405 $
A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, the sum of these four angles, expressed in degrees, is:
$ \textbf{(A)}\ 1080\qquad
\textbf{(B)}\ 900\qquad
\textbf{(C)}\ 720\qquad
\textbf{(D)}\ 540\qquad
\textbf{(E)}\ 360$
$n$ players participated in a competition. Any two players have played exactly one game, and there was no tie game. For a set of $k(\le n)$ players, if it is able to line the players up so that each player won every player at the back, we call the set [i]ranked[/i]. For each player who participated in the competition, the set of players who lost to the player is ranked. Prove that the whole set of players can be split into three or less ranked sets.
Show that the sum of the (local) maximum and minimum values of the function $\frac{tan(3x)}{tan^3x}$ on the interval $\big(0, \frac{\pi }{2}\big)$ is rational.
If $a_i >0$ ($i=1, 2, \cdots , n$) and $\sum \limits_{i=1}^n a_i^k=1$, where $1\leq k\leq n+1$, then $$\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}$$
Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.
A light car and a heavy truck have the same momentum. The truck weighs ten times as much as the car. How do their kinetic energies compare?
$\textbf{(A)}$ The truck's kinetic energy is larger by a factor of $100$
$\textbf{(B)}$ They truck's kinetic energy is larger by a factor of $10$
$\textbf{(C)}$ They have the same kinetic energy
$\textbf{(D)}$ The car's kinetic energy is larger by a factor of $10$
$\textbf{(E)}$ The car's kinetic energy is larger by a factor of $100$
A $\emph{tetromino tile}$ is a tile that can be formed by gluing together four unit square tiles, edge to edge. For each positive integer $\emph{n}$, consider a bathroom whose floor is in the shape of a $2\times2 n$ rectangle. Let $T_n$ be the number of ways to tile this bathroom floor with tetromino tiles. For example, $T_2 = 4$ since there are four ways to tile a $2\times4$ rectangular bathroom floor with tetromino tiles, as shown below.
[click for diagram]
Prove that each of the numbers $T_1, T_2, T_3, ...$ is a perfect square.
4. There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$?
[i]Proposed by Tragic[/i]
A $1 \times n$ rectangle ($n \geq 1 $) is divided into $n$ unit ($1 \times 1$) squares. Each square of this rectangle is colored red, blue or green. Let $f(n)$ be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of $f(9)/f(3)$? (The number of red squares can be zero.)
There is a table with $n$ rows and $18$ columns. Each of its cells contains a $0$ or a $1$. The table satisfies the following properties:
[list=1]
[*]Every two rows are different.
[*]Each row contains exactly $6$ cells that contain $1$.
[*]For every three rows, there exists a column so that the intersection of the column with the three rows (the three cells) all contain $0$.
[/list]
What is the greatest possible value of $n$?
Let $ v_1, v_2, \ldots, v_{1989}$ be a set of coplanar vectors with $ |v_r| \leq 1$ for $ 1 \leq r \leq 1989.$ Show that it is possible to find $ \epsilon_r$, $1 \leq r \leq 1989,$ each equal to $ \pm 1,$ such that \[ \left | \sum^{1989}_{r\equal{}1} \epsilon_r v_r \right | \leq \sqrt{3}.\]
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself).
Show that $f(1)=0$.
An acute triangle $\bigtriangleup ABC$ is given which $BC>CA>AB$.
$I$ is the interior and the incircle of $\bigtriangleup ABC$ meets $BC, CA, AB$ at $D,E,F$. $AD$ and $BE$ meet at $P$. Let $l_{1}$ be a tangent from D to the circumcircle of $\bigtriangleup DIP$, and define $l_{2}$ and $l_{3}$ on $E$ and $F$, respectively.
Prove $l_{1},l_{2},l_{3}$ meet at one point.
Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.
Akshar is reading a $500$ page book, with odd numbered pages on the left, and even numbered pages on the right. Multiple times in the book, the sum of the digits of the two opened pages are $18$. Find the sum of the page numbers of the last time this occurs.
Proposed by Minseok Eli Park (wolfpack)
Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .
In the space there are given crossed lines $s$ and $t$ such that $\angle(s,t)=60^\circ$ and a segment $AB$ perpendicular to them. On $AB$ it is chosen a point $C$ for which $AC:CB=2:1$ and the points $M$ and $N$ are moving on the lines $s$ and $t$ in such a way that $AM=2BN$. The angle between vectors $\overrightarrow{AM}$ and $\overrightarrow{BM}$ is $60^\circ$. Prove that:
(a) the segment $MN$ is perpendicular to $t$;
(b) the plane $\alpha$, perpendicular to $AB$ in point $C$, intersects the plane $CMN$ on fixed line $\ell$ with given direction in respect to $s$;
(c) all planes passing by $ell$ and perpendicular to $AB$ intersect the lines $s$ and $t$ respectively at points $M$ and $N$ for which $AM=2BN$ and $MN\perp t$.
Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$.
Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.
Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that
\[\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r. \]
[i]Alternative formulation.[/i] In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$.
Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$.
[i]Proposed by Dirk Laurie, South Africa[/i]
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
Prove the following inequality.
\[ \sum_{n=0}^\infty \int_0^1 x^{4011} (1-x^{2006})^\frac{n-1}{2006}\ dx<\frac{2006}{2005} \]