Found problems: 85335
Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining
vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle.
Let this new rectangle be given. Restore the original rectangle using compass and ruler.
How many positive integers less than 10,000 have at most two different digits?
The solutions to the system of equations
\begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*}
are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$
Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$, such that $ (2 \minus{} \sin 2x)\sin (x \plus{} \frac {\pi}{4}) \equal{} 1$.
Given a square side $1$ and $2n$ positive reals $a_1, b_1, ... , a_n, b_n$ each $\le 1$ and satisfying $\sum a_ib_i \ge 100$. Show that the square can be covered with rectangles $R_i$ with sides length $(a_i, b_i)$ parallel to the square sides.
Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions:
[list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$
[*] For all real number $x$, $f(g(x))=g(f(x))=0$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$
[/list]
In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.
The polynomial $ P(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $ y$-intercept of the graph of $ y \equal{} P(x)$ is 2, what is $ b$?
$ \textbf{(A)} \ \minus{} 11 \qquad \textbf{(B)} \ \minus{} 10 \qquad \textbf{(C)} \ \minus{} 9 \qquad \textbf{(D)} \ 1 \qquad \textbf{(E)} \ 5$
Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.
Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that
\[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \]
[i]Radu Miculescu[/i]
Given three colors and a rectangle m × n dice unit, we want to color each segment constituting one side of a square drive with one of three colors so that each square unit have two sides of one color and two sides another color. How many colorings we have?
Prove that there are only [b]finitely[/b] positive integer $a$ such that $a-2006=\sum\limits_{i=1}^{2006} 2^ia_i$ with $\{a_i\}$ as divisors (not necessary distinct) of $n$.
Find all $t\in \mathbb Z$ such that: exists a function $f:\mathbb Z^+\to \mathbb Z$ such that:
$f(1997)=1998$
$\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$
For how many positive integers $ n$ is $ n^2\minus{}3n\plus{}2$ a prime number?
$ \textbf{(A)}\ \text{none} \qquad
\textbf{(B)}\ \text{one} \qquad
\textbf{(C)}\ \text{two} \qquad
\textbf{(D)}\ \text{more than two, but finitely many}\\
\textbf{(E)}\ \text{infinitely many}$
Let $F$ be the set of all $n$-tuples $(A_1, \ldots, A_n)$ such that each $A_{i}$ is a subset of $\{1, 2, \ldots, 1998\}$. Let $|A|$ denote the number of elements of the set $A$. Find
\[ \sum_{(A_1, \ldots, A_n)\in F} |A_1\cup A_2\cup \cdots \cup A_n| \]
To prepare a stay abroad, students meet at a workshop including an excursion. To promote interaction between the students, they are to be distributed to two busses such that not too many of the students in the same bus know each other.
Every student knows all those who know her. The number of such pairwise acquaintances is $k$.
Prove that it is possible to distribute the students such that the number of pairwise acquaintances in each bus is at most $\frac{k}{3}$.
Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$.
(a) Let $p$ be a prime. Consider the graph whose vertices are the ordered pairs $(x, y)$ with $x, y \in\{0, 1, . . . , p-1\}$ and whose edges join vertices $(x, y)$ and $(x' , y')$ if and only if $xx'+yy'\equiv 1 \pmod{p}$ . Prove that this graph does not contain $C_{4}$ .
(b) Prove that for infinitely many values $n$ there is a graph $G_{n}$ with $e(G_{n}) \geq \frac{n\sqrt{n}}{2}-n$ that does not contain $C_{4}$.
Let $\triangle{ABC}$ be a triangle, $\omega$ its circumcircle and $O$ the center of $\omega$. Let $P$ be a point on the segment $BC$. We denote by $Q$ the second intersection point of the circumcircles of triangles $\triangle{AOB}$ and $\triangle{APC}$. Prove that the line $PQ$ and the tangent to $\omega$ at point $A$ intersect on the circumcircle of triangle $\triangle AOB$.
Draw the graph of the function $y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |$.
For positive integers $t,a$, and $b$, Lucy and Windy play the $(t,a,b)$- [i]game [/i] defined by the following rules. Initially, the number $t$ is written on a blackboard. On her turn, a player erases the number on the board and writes either the number $t - a$ or $t - b$ on the board. Lucy goes first and then the players alternate. The player who first reaches a negative losses the game. Prove that there exist infinitely many values of $t$ in which Lucy has a winning strategy for all pairs $(a, b)$ with $a + b = 2005$.
Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.
Let $A$ be the number of all points with integer coordinates in a three-dimensional coordinate system. We assume that nine arbitrary points in $A$ will be colored blue. Show that we can always find two blue dots so that the line segment between them contains at least one point from $A$.
Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. Prove that if $AP=4PN$, then $ABCD$ is a parallelogram.
Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$?
$ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$
Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$