Found problems: 85335
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
In an acute angled triangle $ABC$ , let $BB' $ and $CC'$ be the altitudes. Ray $C'B'$ intersects the circumcircle at $B''$ andl let $\alpha_A$ be the angle $\widehat{ABB''}$. Similarly are defined the angles $\alpha_B$ and $\alpha_C$. Prove that $$\displaystyle\sin \alpha _A \sin \alpha _B \sin \alpha _C\leq \frac{3\sqrt{6}}{32}$$
(Romania)
Let $I$ be the incenter of an acute-angled triangle $ABC$. Let $P$, $Q$, $R$ be points on sides $AB$, $BC$, $CA$ respectively, such that $AP=AR$, $BP=BQ$ and $\angle PIQ = \angle BAC$. Prove that $QR \perp AC$.
How many permutations $a_1, a_2, \ldots, a_n$ of $\{1, 2, . . ., n \}$ are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange $a_i$ and $a_{i+1}$ whenever $a_i > a_{i+1}$ for $i$ running from $1$ up to $n - 1 \ ?$
For every nonempty subset $X$ of $M=\{1,2,\ldots,2000\}$, $a_X$ denotes the sum of the minimum and maximum element of $X$. Compute the arithmetic mean of the numbers $a_X$ when $X$ goes over all nonempty subsets $X$ of $M$.
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
Given a cube with side length $ 1$, we perform six cuts as follows: one cut parallel to the $xy$-plane, two cuts parallel to the $yz$-plane, and three cuts parallel to the $xz$-plane, where the cuts are made uniformly independent of each other. What is the expected value of the volume of the largest piece?
The incircle of the triangle $ABC$ touches the sides $AC$ and $BC$ at points $K$ and $L$, respectively. the $B$-excircle touches the side $AC$ of this triangle at point $P$. Line $KL$ intersects with the line passing through $A$ and parallel to $BC$ at point $M$. Prove that $PL = PM$.
A regular dodecagon $A_1A_2...A_{12}$ is inscribed in a circle with a diameter of $20$ cm . Calculate the perimeter of the pentagon $A_1A_3A_6A_8A_{11}$.
(Alexey Panasenko)
Two flag poles of height $11$ and $13$ are planted vertically in level ground, and an equilateral triangle is hung as shown in the figure so that [the] lowest vertex just touches the ground. What is the length of the side of the equilateral triangle?
[asy]
pair A, B, C, D, E;
A = (0,11);
B = origin;
C = (13.8564064606,0);
D = (13.8564064606,13);
E = (8.66025403784,0);
draw(A--B--C--D--cycle);
draw(A--E--D);
label("$11$",A--B);
label("$13$",C--D);
[/asy]
Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively.
Prove that
$$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$
where $\sigma(P)$ denotes the area of a polygon $P.$
Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that
$$\frac{a-2}{a+1}+\frac{b-2}{b+1}+\frac{c-2}{c+1} \le 0$$
For real numbers $a,b,c$ and positive number $\lambda$ such that three real roots $x_1,x_2,x_3$ of $f(x)=x^3+ax^2+bx+c$ satisfying:
$(1) x_2-x_1=\lambda$;
$(2) x_3>\frac{1}{2}(x_1+x_2)$.
Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$
Find all polynomials $P(X)$ with real coefficients such that if real numbers $x,y$ and $z$ satisfy $x+y+z=0,$ then the points $\left(x,P(x)\right), \left(y,P(y)\right), \left(z,P(z)\right)$ are all colinear.
In $ n$-dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue.
[i]A. Csaszar[/i]
A factorization of a positive integers is a way of writing it as a product of positive integers greater than $1$. Two factorizations are considered the same if they only differ in the order of terms in the product. For instance, $18$ has $4$ different factorizations: $18, 2\cdot 9, 3\cdot 6$ and $ 2\cdot 3\cdot 3$. For a positive integer $n$ we denote by $f(n)$ the number of distinct factorizations of $n$. By convention $f(1)=1$. Prove that $f(n)\leq n$ for all positive integers $n$.
Let $m,n \geq 2$ be integers with gcd$(m,n-1) = $gcd$(m,n) = 1$. Prove that among $a_1, a_2, \ldots, a_{m-1}$, where $a_1 = mn+1, a_{k+1} = na_k + 1$, there is at least one composite number.
$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.
[i]Proposed by Yang Liu[/i]
Consider the sequence formed by the first digits of the powers of $5$:$$1,5,2,1,6,...$$ Prove any segment in this sequence, when written in reversed order, will be encountered in the sequence of the first digits of the powers of $2:$
$$1,2,4,8,1,3,6,1...$$
In a parallelogram $ABCD$, the bisector of $\angle A$ intersects $BC$ at $M$ and the extension of $DC$ at $N$. Let $O$ be the circumcircle of the triangle $MCN$. Prove that $\angle OBC = \angle ODC$
Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.)
[asy]
unitsize(18);
draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle);
draw((0,1)--(3,1));
draw((2,0)--(2,3));
draw((1,1)--(1,3));
label("A",(0.5,2));
label("B",(1.5,2));
label("C",(2.5,2));
label("D",(1,0.5));
[/asy]
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $
Prove that the equation \[6(6a^{2}+3b^{2}+c^{2}) = 5n^{2}\] has no solutions in integers except $a=b=c=n=0$.
Find all the non-zero polynomials $P(x),Q(x)$ with real coefficients and the minimum degree,such that for all $x \in \mathbb{R}$:
\[ P(x^2)+Q(x)=P(x)+x^5Q(x) \]