Found problems: 85335
Find all integers $n \ge 4$ with the following property:
Each field of the $n \times n$ table can be painted white or black in such a way that each square of this table had the same color as exactly the two adjacent squares. (Squares are adjacent if they have exactly one side in common.)
How many different colorings of the $6 \times 6$ table fields meet the above conditions?
$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.
As $x$ ranges over the interval $(0,\infty)$, the function \[\sqrt{9x^2 + 173x + 900} - \sqrt{9x^2 + 77x + 900}\] ranges over the interval $(0,M)$. Find $M$.
In $\triangle ABC$, if $9a^2+9b^2-19c^2=0$, then $\frac{\cot C}{\cot A+\cot B}=$________.
A committee with $20$ members votes for the candidates $A,B,C$ by a different election system. Each member writes his ordered prefer list to the ballot (e.g. if he writes $BAC$, he prefers $B$ to $A$ and $C$, and prefers $A$ to $C$). After the ballots are counted, it is recognized that each of the six different permutations of three candidates appears in at least one ballot, and $11$ members prefer $A$ to $B$, $12$ members prefer $C$ to $A$, $14$ members prefer $B$ to $C$. How many members are there such that $B$ is the first choice of them?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{More information is needed}
$
Let $S$ be a nonempty set of primes satisfying the property that for each proper subset $P$ of $S$, all the prime factors of the number $\left(\prod_{p\in P}p\right)-1$ are also in $S$. Determine all possible such sets $S$.
$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic. [i](20 Points)[/i]
What is the smallest integer $m$ such that $\frac{10!}{m}$ is a perfect square?
$\text{(A) }2\qquad\text{(B) }7\qquad\text{(C) }14\qquad\text{(D) }21\qquad\text{(E) }35$
Given the set of numbers $ \{1, 2, 3, \ldots, 100\} $. From this set, create 10 pairwise disjoint subsets $ N_i = \{a_{i,1}, a_{i,2}, ... a_{i,10} $ ($ i = 1, 2, \ldots, 10 $ ) so that the sum of the products
$$
\sum_{i=10}^{10}\prod_{j=1}^{10} a_{i,j}
$$
was the biggest.
For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.
Given a trapezoid, divide it by a line into two quadrilaterals in such a way that both of them are cyclic with the same circumradius. Discuss conditions of solvability.
Let $ n$ be a natural number $ \ge 2$ which divides $ 3^n\plus{}4^n$.Prove That $ 7\mid n$.
Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number.
Prove that at least $2^{n-3}$ subsets of $X$ are compact.
[i]Proposed by Mahyar Sefidgaran[/i]
Prove the inequality
\[(-a+b+c)^2(a-b+c)^2(a+b-c)^2 \geq (-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2)\]
for all real numbers $a, b, c.$
Yesterday (=April 22, 2003) was Gittes birthday. She notices that her age equals the sum of the 4 digits of the year she was born in.
How old is she?
Prove that a right triangle has an angle equal to $30^o$ if and only if the center of the circle inscribed in this triangle is located on the perpendicular bisector of the median taken from the vertex of the right angle of the triangle.
Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $
[i]Gheorghe Iurea[/i]
A trapezium $ ABCD$, in which $ AB$ is parallel to $ CD$, is inscribed in a circle with centre $ O$. Suppose the diagonals $ AC$ and $ BD$ of the trapezium intersect at $ M$, and $ OM \equal{} 2$.
[b](a)[/b] If $ \angle AMB$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides.
[b](b)[/b] If $ \angle AMD$ is $ 60^\circ ,$ find, with proof, the difference between the lengths of the parallel sides.
[b][Weightage 17/100][/b]
Determine all polynomials P(x) with real coefficients such that
[(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.
Let $G$ be the set of polynomials of the form
\[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\]
where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
${ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D}}\ 992\qquad\textbf{(E)}\ 1056 $
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.
$(NET 5)$ The bisectors of the exterior angles of a pentagon $B_1B_2B_3B_4B_5$ form another pentagon $A_1A_2A_3A_4A_5.$ Construct $B_1B_2B_3B_4B_5$ from the given pentagon $A_1A_2A_3A_4A_5.$
Let $n$ be a positive even integer, and let $c_1, c_2, \dots, c_{n-1}$ be real numbers satisfying \[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \] Prove that \[
2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2
\] has no real roots.
Let $m,n$ be two positive integers with $m \ge n \ge 2022$. Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be $2n$ real numbers. Prove that the numbers of ordered pairs $(i,j) ~(1 \le i,j \le n)$ such that
\[ |a_i+b_j-ij| \le m \]
does not exceed $3n\sqrt{m \log n}$.
Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c,B_a, B_c, C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be?
(A. Garkavyj, A. Sokolov )