Found problems: 85335
Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.
Inside an angle $\angle BOC$ there are three disjoint circles: $k_1,k_2$ and $k_3$, which are, each one, tangent to its sides $BO$ and $OC$. Let $r_1, r_2$ and $r_3$, respectively, be the radii of these circles, with $r_1<r_2<r_3$. The circles $k_1$ and $k_3$ are tangent to the side $OB$ at $A$ and $B$, respectively, and $k_2$ is tangent to the side $OC$ at $C$. Let $K=AC\cap k_1,L=AC\cap k_2,M=BC\cap k_2$ and $N=BC\cap k_3$. Besides that, let $P=AM\cap BK,Q=AM\cap BL,R=AN\cap BK$ and $S=AN\cap BL$. If the intersections of $CP,CQ,CR$ and $CS$ with $AB$ are $X,Y,Z$ and $T$, respectively, prove that $XZ = YT$.
In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that
a) $AC^2 = AB^2 + AB \cdot BC$;
b) $AB+BC < 2 \cdot AC$.
$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?
Find all real solutions to $x^4+(2-x)^4=34$.
A board $ n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $O$ to $n$ included, are written down: one integer in each unit square, in such a way that the sums of integers in each $2\times 2$ square of the board are different. Find all $n$ for which such boards exist.
Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ?
Mikhail Evdokimov
Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$,
$$a_{ij}=\begin{cases}
(-1)^{|i-j|} & \text{if}\, i\ne j,\\
2 & \text{if}\, i= j.
\end{cases}$$
Prove that if $ n \geq 2 $ and $ I_ {1}, I_ {2}, \ldots, I_ {n} $ are idealized in a unit-element commutative ring such that any nonempty $ H \subseteq \{ 1,2, \dots, n \} $ then if $ \sum_ {h \in H} I_ {h} $ Is ideal
$$
I_ {2} I_ {3} I_ {4} \dots I_ {n} + I_ {1} I_ {3} I_ {4} \dots I_ {n} + \dots + I_ {1} I_ {2} \dots I_ {n-1}
$$also Is ideal.
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.
From eight edges and eight diagonal of surfaces of a cube, choose $k$ lines. If any two lines of them are skew lines, then the maximum value of $k$ is________.
[b]p1.[/b] Darth Vader urgently needed a new Death Star battle station. He sent requests to four planets asking how much time they would need to build it. The Mandalorians answered that they can build it in one year, the Sorganians in one and a half year, the Nevarroins in two years, and the Klatoonians in three years. To expedite the work Darth Vader decided to hire all of them to work together. The Rebels need to know when the Death Star is operational. Can you help the Rebels and find the number of days needed if all four planets work together? We assume that one year $= 365$ days.
[b]p2.[/b] Solve the inequality: $\left( \sin \frac{\pi}{12} \right)^{\sqrt{1-x}} > \left( \sin \frac{\pi}{12} \right)^x$
[b]p3.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 3x - 6$
[b]p4.[/b] Solve the system of inequalities on $[0, 2\pi]$:
$$\sin (2x) \ge \sin (x)$$
$$\cos (2x) \le \cos (x)$$
[b]p5.[/b] The planet Naboo is under attack by the imperial forces. Three rebellian camps are located at the vertices of a triangle. The roads connecting the camps are along the sides of the triangle. The length of the first road is less than or equal to $20$ miles, the length of the second road is less than or equal to $30$ miles, and the length of the third road is less than or equal to $45$ miles. The Rebels have to cover the area of this triangle by a defensive field. What is the maximal area that they may need to cover?
[b]p6.[/b] The Lake Country on the planet Naboo has the shape of a square. There are nine roads in the country. Each of the roads is a straight line that divides the country into two trapezoidal parts such that the ratio of the areas of these parts is $2:5$. Prove that at least three of these roads intersect at one point.
PS. You should use hide for answers.
$S=\{1,2, \ldots, 6\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \subseteq S$ and $A \cap B=\phi$
[list=1]
[*] 360
[*] 364
[*] 365
[*] 366
[/list]
Let $x, y$ be positive integers such that
$$
x^4=(x-1)\left(y^3-23\right)-1 .
$$
Find the maximum possible value of $x+y$.
Vijay has a stash of different size stones: in particular, he has $2021$ types of stones, with sizes from $0$ through $2020$, and he has $2r+1$ stones of size $r$.
Vijay starts randomly (and without replacement) taking out stones from his stash and laying them out in a line. Vijay notices that the first stone of size $2020$ comes before the first stone of size $2019$, the first stone of size $2019$ is before the first stone of size $2018$, and so on. What is the probability of this happening?
Express your answer in terms of only basic arithmetic operations (division, exponentiation, etc.) and the factorial function.
[i]Proposed by Misha Ivkov[/i]
Let $n$ and $r$ two positive integers. It is wanted to make $r$ subsets $A_1,\ A_2,\dots,A_r$ from the set $\{0,1,\cdots,n-1\}$ such that all those subsets contain exactly $k$ elements and such that, for all integer $x$ with $0\leq{x}\leq{n-1}$ there exist $x_1\in{}A_1,\ x_2\in{}A_2 \dots,x_r\in{}A_r$ (an element of each set) with $x=x_1+x_2+\cdots+x_r$.
Find the minimum value of $k$ in terms of $n$ and $r$.
The real numbers $a, b, x$ satisfy the inequalities $| a + x + b | \le 1, | 4a + 2x + b | \le1, | 9a + 6x + 4b | \le 1$.
Prove that $| x | \le15$.
The chemical potential energy stored in a battery is converted into kinetic energy in a toy car that increases its speed first from $0 \text{ mph}$ to $2 \text{ mph}$ and then from $2 \text{ mph}$ up to $4 \text{ mph}$. Ignore the energy transferred to thermal energy due to friction and air resistance. Compared to the energy required to go from $0$ to $2 \text{ mph}$, the energy required to go from $2$ to $4 \text{ mph}$ is
$ \textbf{(A)}\ \text{half the amount.}$
$ \textbf{(B)}\ \text{the same amount.}$
$ \textbf{(C)}\ \text{twice the amount.}$
$ \textbf{(D)}\ \text{three times the amount.}$
$ \textbf{(E)}\ \text{four times the amount.} $
A snail begins a journey starting at the origin of a coordinate plane. The snail moves along line segments of length $\sqrt{10}$ and in any direction such that the horizontal and vertical displacements are both integers. As the snail moves, it leaves a trail tracing out its entire journey. After a while, this trail can form various polygons. What is the smallest possible area of a polygon that could be created by the snail's trail?
Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $
You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?
In a convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ are equal. We construct four equilateral triangles with centers $O_1,O_2,O_3,O_4$ on the sides sides $AB, BC, CD, DA$ outside of this quadrilateral, respectively. Show that $O_1O_3 \perp O_2O_4$.
Three points $P_1, P_2,$ and $P_3$ and three lines $\ell_1, \ell_2,$ and $\ell_3$ lie in the plane such that none of the three points lie on any of the three lines. For (not necessarily distinct) integers $i$ and $j$ between 1 and 3 inclusive, we call a line $\ell$ $(i, j)$-[i]good[/i] if the reflection of $P_i$ across $\ell$ lies on $\ell_j$, and call it [i]excellent[/i] if there are two distinct pairs $(i_1, j_1)$ and $(i_2, j_2)$ for which it is good. Suppose that exactly $N$ excellent lines exist. Compute the largest possible value of $N$.
[i]Proposed by Yannick Yao[/i]
For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$