Found problems: 85335
On March $6$, $2002$, the celebrations of the $500$th anniversary of the birth of by mathematician Pedro Nunes. That morning, only ten people entered the Viva bookstore for science. Each of these people bought exactly $3$ different books. Furthermore, any two people bought at least one copy of the same book. The Adventures of Mathematics by Pedro Nunes was one of the books that achieved the highest number of sales in this morning. What is the smallest value this number could have taken?
Let $\{a_1,...,a_n\}\subset \{-1,1\}$ and $a>0$ . Denote by $X$ and $Y$ the number of collections $\{\varepsilon_1,...,\varepsilon_n\}\subset \{-1,1\}$, such that $$max_{1\le k\le n}(\varepsilon_1a_1+...+\varepsilon_ka_k) >\alpha$$ and $$\varepsilon_1a_1+...+\varepsilon_na_n>a$$ respectively. Prove that $X\le 2Y$.
Let $n$ be a positive integer. Show that \[\sum^{n}_{k=1}\tan^{2}\frac{k \pi}{2n+1}\] is an odd integer.
A vertical mass-spring oscillator is displaced $\text{2.0 cm}$ from equilibrium. The $\text{100 g}$ mass passes through the equilibrium point with a speed of $\text{0.75 m/s}$. What is the spring constant of the spring?
(A) $\text{90 N/m}$
(B) $\text{100 N/m}$
(C) $\text{110 N/m}$
(D) $\text{140 N/m}$
(E) $\text{160 N/m}$
Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$.
(a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle.
(b) Prove that the line $ST$ is parallel to the side $BC$.
(Karl Czakler)
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.)
[i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]
Let $A$ be a finite set of points in the coordinate plane. Suppose that $A$ has $n\geq3$ points. Given any $a$ in $A$, the horizontal and vertical lines through $a$ define four [i]closed[/i] quadrants centered at $a$. For any real number $\alpha$, call a point $a$ in $A$ $\alpha$-good if there are two diagonally opposite closed quadrants centered at $a$ that each contain at least $\alpha n$ points from $A$. Show that there is some $a$ in $A$ that is $\frac{1}{8}$-good.
Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that
\[f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.\]
Let $g$ be a function for which
\[g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.\]
Prove that $g$ is bounded.
Find all pairs $(a,b)$ of real numbers such that
$$|\sqrt{1-x^2 }-ax-b| \leq \frac{\sqrt{2} -1}{2}$$
holds for all $x\in [0,1]$.
A Rhombus and an Isosceles trapezoid that has same area is drawn in the same circle's outside. Compare their acute angles \\
(explain your answer)
The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$.
\[ \begin{tabular}{|c|c|c|}\hline A & 1 & B \\ \hline 5 & C & 13\\ \hline D & E & 3 \\ \hline\end{tabular} \]
$\textbf{(A)}\ 32 \qquad \textbf{(B)}\ 28 \qquad \textbf{(C)}\ 26 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 16$
Prove that for every prime number $p$, there are infinitely many positive integers $n$ such that $p$ divides $2^n - n$.
Find the flux of the vector field $\overrightarrow{r}/r^3$ through the surface
\[(x-1)^2+y^2+z^2=2\]
A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?
Each of the numbers in the set $N = \{1, 2, 3, \cdots, n - 1\}$, where $n \geq 3$, is colored with one of two colors, say red or black, so that:
[i](i)[/i] $i$ and $n - i$ always receive the same color, and
[i](ii)[/i] for some $j \in N$, relatively prime to $n$, $i$ and $|j - i|$ receive the same color for all $i \in N, i \neq j.$
Prove that all numbers in $N$ must receive the same color.
Let n be a positive integer and let $x_1, x_2,...,x_n$ and $y_1, y_2,...,y_n$ be real numbers. Prove that there exists a number $i, i = 1, 2,...,n$, such that $$\sum_{j=1}^n |x_i - x_j | \le \sum_{j=1}^n |x_i - y_j | $$
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.
[img]http://up.iranblog.com/Files7/dda310bab8b6455f90ce.jpg[/img]
A $(0_x, 1_y, 2_z)$-string is an infinite ternary string such that:
[list]
[*] If there is a $0$ in position $i$ then there is a $1$ in position $i + x$,
[*] if there is a $1$ in position $j$ then there is a $2$ in position $j + y$,
[*] if there is a $2$ in position $k$ then there is a $0$ in position $k + z$.
[/list]
For how many ordered triples of positive integers $(x, y, z)$ with $x, y, z \leq 100$ does there exist $(0_x, 1_y, 2_z)$-string?
An arbitary point $D$ is selected on arc $BC$ not containing $A$ on $(ABC)$. $P$ and $Q$ are the reflections of point $B$ and $C$ with respect to $AD$, respectively. Circumcircles of $ABQ$ and $ACP$ intersect at $E\neq A$. Prove that $A;D;E$ is colinear
A string of letters is called $good$ if it contains a continuous substring $IMONST$ in it. For example, the string $NSIMONSTIM$ is $good$, but the string $IMONNNST$ is not.
Find the number of good strings consisting of $12$ letters from $I$, $M$, $O$, $N$, $S$, $T$ only.
Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$.
a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$.
b) Prove that $A, K, L$ are collinear.
Dürer's $n \times m$ garden is surgically divided into $n \times m$ unit squares, and in the middle of one of these squares, he planted his favourite petunia. Dürer's gardener struggles with a mole, trying to drive him out of the magnificent garden, so he builds an underground wall on the edge of the garden. The only problem is that the mole managed to stay inside the walls.. When the mole meets a wall, it changes it's direction as if it was "reflected", that is, proceeding his route in the direction that includes the same angle with the wall as his direction before. The mole starts beneath the petunia, in a direction that includes a $45^o$ angle with the walls. Is it possible for the mole to cross the petunia in a direction perpendicular to it's original direction?
(Think in terms of $n,m$.)
Let $ABC$ be a triangle. Construct three circles $k_1$, $k_2$, and $k_3$ with the same radius such that they intersect each other at a common point $O$ inside the triangle $ABC$ and $k_1\cap k_2=\{A,O\}$, $k_2 \cap k_3=\{B,O\}$, $k_3\cap k_1=\{C,O\}$. Let $t_a$ be a common tangent of circles $k_1$ and $k_2$ such that $A$ is closer to $t_a$ than $O$. Define $t_b$ and $t_c$ similarly. Those three tangents determine a triangle $MNP$ such that the triangle $ABC$ is inside the triangle $MNP$. Prove that the area of $MNP$ is at least $9$ times the area of $ABC$.
In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.