Found problems: 260
Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.
Find all integer triples $(m,p,q)$ satisfying \[2^mp^2+1=q^5\] where $m>0$ and both $p$ and $q$ are prime numbers.
Let $ X$ be the set of all positive integers greater than or equal to $ 8$ and let $ f: X\rightarrow X$ be a function such that $ f(x\plus{}y)\equal{}f(xy)$ for all $ x\ge 4, y\ge 4 .$ if $ f(8)\equal{}9$, determine $ f(9) .$
Each point of a plane is painted in one of three colors. Show that there exists a triangle such that:
$ (i)$ all three vertices of the triangle are of the same color;
$ (ii)$ the radius of the circumcircle of the triangle is $ 2008$;
$ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.
Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.
Let $ G$ be an infinite compact topological group with a Hausdorff topology. Prove that $ G$ contains an element $ g \not\equal{} 1$ such that the set of all powers of $ g$ is either everywhere dense in $ G$ or nowhere dense in $ G$.
[i]J. Erdos[/i]
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is,
\[N=1111\cdots 11.\]
Find the thousandth digit after the decimal point of $\sqrt N$.
Let $ a,b,c$ be nonnegative real numbers. Prove that:
$ \frac{1}{3}((a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2) \le a^2\plus{}b^2\plus{}c^2\minus{}3 \sqrt[3]{a^2 b^2 c^2 } \le (a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2.$
How many positive integers not exceeding $ 2001$ are multiples of $ 3$ or $ 4$ but not $ 5$?
$ \textbf{(A)}\ 768 \qquad
\textbf{(B)}\ 801 \qquad
\textbf{(C)}\ 934 \qquad
\textbf{(D)}\ 1067 \qquad
\textbf{(E)}\ 1167$
Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that
\[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]
Some $ m$ squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest $ m$ for which it is possible to mark all squares after several such operations.
Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.
If $ f$ is a nonnegative, continuous, concave function on the closed interval $ [0,1]$ such that $ f(0)=1$, then \[ \int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command.
\int_0^1 f(x)dx \right]^2.\]
[i]Z. Daroczy[/i]
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:
\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]
When does equality occur?
Find the last three digits of the product of the positive roots of \[ \sqrt{1995}x^{\log_{1995}x}=x^2. \]
Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.
Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .
Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that
a) $ 0$ is the only nilpotent element of $ A$;
b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.
For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.
There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. Which of the following is it?
$ \text{(A)}\ 5724\qquad\text{(B)}\ 7245\qquad\text{(C)}\ 7254\qquad\text{(D)}\ 7425\qquad\text{(E)}\ 7542 $
Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given.
i) Prove that if $ a_1>2$, then:
\[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\]
ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.
Let $a$, $b$, $n$ be positive integers with $a,b > 1$ and $\gcd(a,b) = 1$. Prove that $n$ divides $\phi\left(a^{n}+b^{n}\right)$.
Prove that if $a,b,c$ are positive integers such that $c^2 = a^2+b^2$, then both $c^2+ab$ and $c^2-ab$ are also expressible as the sums of squares of two positive integers.
A finite set of points $P$ in the plane has the following property: Every line through two points in $P$ contains at least one more point belonging to $P$. Prove that all points in $P$ lie on a straight line.
[hide="Remark."]This may be a well known theorem called "Sylvester Gallai", but I didn't find this problem (I mean, exactly this one) using search function. So please discuss about the problem here, in this topic. Thanks :) [/hide]
An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?
$ \textbf{(A)}\ 25\left(\frac{2}{3}\right)^{50}\qquad\textbf{(B)}\ \frac{1}{2}\left(1 - \frac{1}{3^{50}}\right)\qquad\textbf{(C)}\ \frac{1}{2}\qquad\textbf{(D)}\ \frac{1}{2}\left(1 + \frac{1}{3^{50}}\right)\qquad\textbf{(E)}\ \frac{2}{3} $