Found problems: 85335
Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs.
a) Determine the maximum possible area of a trio.
b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon.
c) Determine the total number of triangles associated with the hexagon.
Find the equation of cylinder that passes three straight lines
$L_1=
\begin{cases}
x=0\\
y-z=2
\end{cases},
L_2=
\begin{cases}
x=0\\
x+y-z+2=0
\end{cases},
L_3=
\begin{cases}
x=\sqrt{2}\\
y-z=0
\end{cases}$.
Let the sequence $(a_{n})_{n\geqslant 1}$ be defined as:
$$a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}},$$
where $A_{m}^{k}$ are defined by $$A_{m}^{k}=\binom{m}{k}\cdot k!.$$
Prove that $$a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}.$$
Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.
The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.
$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$
Let $f:[0, \infty) \to \mathbb{R}$ a continuous function, constant on $\mathbb{Z}_{\geq 0}.$ For any $0 \leq a < b < c < d$ which satisfy $f(a)=f(c)$ and $f(b)=f(d)$ we also have $f \left( \frac{a+b}{2} \right) = f \left( \frac{c+d}{2} \right).$
Prove that $f$ is constant.
A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$.
[i]Albania[/i]
Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.
Compute the greatest common divisor of $4^8 - 1$ and $8^{12} - 1$.
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
Find all integers $n$ such that if
\[
1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n
\]
are the divisors of $n$, then the sequence
\[
d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1}
\]
forms a permutation of an arithmetic progression.
[i](Kritesh Dhakal, Nepal)[/i]
Suppose there exists a group with exactly $n$ subgroups of index 2. Prove that there exists a finite abelian group $G$ that has exactly $n$ subgroups of index 2.
Define a sequence by $a_0=2019$ and $a_n=a_{n-1}^{2019}$ for all positive integers $n$. Compute the remainder when \[a_0+a_1+a_2+\dots+a_{51}\] is divided by $856$.
[i]Proposed by Tristan Shin[/i]
$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as
\[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|.
\]
$ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.
Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]
Let $\triangle ABC$ be a triangle with $AB = 7$, $BC = 8$, and $AC = 9$. Point $D$ is on side $\overline{AC}$ such that $\angle CBD$ has measure $45^\circ$. What is the length of $\overline{BD}$?
Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations:
$$x+y=z^2+w^2+6zw$$
$$x+z=y^2+w^2+6yw$$
$$x+w=y^2+z^2+6yz$$
$$y+z=x^2+w^2+6xw$$
$$y+w=x^2+z^2+6xz$$
$$z+w=x^2+y^2+6xy$$
Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size
$1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What
is the minimum number of cuts that Geetha needs?
Note: After every cut, she can rearrange the pieces before cutting again. At every
cut, she can cut more than one pieces as long as the pieces are on a straight line.
Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length
$\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$
[size=70]*A polygon is called “simple” if it is not self intersecting.[/size]
(a) Find all real numbers $p$ for which the inequality
$$x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)$$
is true for all real numbers $x_1,x_2,x_3$.
(b) Find all real numbers $q$ for which the inequality
$$x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)$$
is true for all real numbers $x_1,x_2,x_3,x_4$.
[i]I. Tonov[/i]
Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$
Let $a,b,c,x,y,z$ be six positive real numbers satisfying $x+y+z=a+b+c$ and $xyz=abc.$ Further, suppose that $a\leq x<y<z\leq c$ and $a<b<c.$ Prove that $a=x,b=y$ and $c=z.$
On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?
$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.