Found problems: 85335
Let $(a_0,a_1,a_2,...)$ and $(b_0,b_1,b_2,...)$ be such sequences of non-negative real numbers, that for every integer $i\geqslant 1$ holds $a_i^2\leqslant a_{i-1}a_{i+1}$ and $b_i^2\leqslant b_{i-1}b_{i+1}$.
Define sequence $c_0,c_1,c_2,...$ as
$$c_0=a_0b_0, \; c_n=\sum_{i=0}^{n} {{n}\choose{i}} a_ib_{n-i}.$$
Prove that for every integer $k\geqslant 1$ holds $c_{k}^2\leqslant c_{k-1}c_{k+1}$.
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies $ 75\%$ of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?
$ \textbf{(A)}\ 2: 1 \qquad
\textbf{(B)}\ 3: 1 \qquad
\textbf{(C)}\ 4: 1 \qquad
\textbf{(D)}\ 16: 3 \qquad
\textbf{(E)}\ 6: 1$
For all positive integers $n$, let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.
Let $n \geq 3$ be an odd number and suppose that each square in a $n \times n$ chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares $a,b$ are considered connected if there exists a sequence of squares $c_1,\ldots,c_k$ with $c_1 = a, c_k = b$ such that $c_i, c_{i+1}$ are adjacent for $i=1,2,\ldots,k-1$.
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Find the maximal number $M$ such that there exists a coloring admitting $M$ pairwise disconnected squares.
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.
A set $A_1 , A_2 , A_3 , A_4$ of 4 points in the plane is said to be [i]Athenian[/i] set if there is a point $P$ of the plane satsifying
(*) $P$ does not lie on any of the lines $A_i A_j$ for $1 \leq i < j \leq 4$;
(**) the line joining $P$ to the mid-point of the line $A_i A_j$ is perpendicular to the line joining $P$ to the mid-point of $A_k A_l$, $i,j,k,l$ being distinct.
(a) Find all [i]Athenian[/i] sets in the plane.
(b) For a given [i]Athenian[/i] set, find the set of all points $P$ in the plane satisfying (*) and (**)
Find the remainder when $5^{2012}$ is divided by $3$.
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that
$$|f(A)\cap f(B)|=|A\cap B|$$
whenever $A$ and $B$ are two distinct subsets of $X$.
[i] (Sergiu Novac)[/i]
A matrix $\begin{bmatrix}
x & y \\
z & w
\end{bmatrix}$ has square root $\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$ if
$$\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}^2
=
\begin{bmatrix}
a^2 + bc &ab + bd \\
ac + cd & bc + d^2
\end{bmatrix}
=
\begin{bmatrix}
x & y \\
z & w
\end{bmatrix}$$
Determine how many square roots the matrix $\begin{bmatrix}
2 & 2 \\
3 & 4
\end{bmatrix}$ has (complex coefficients are allowed).
A triple $(x,y,z)$ of real numbers is called a [i]superparticular[/i] if
$$\frac{x+1}{x} \cdot \frac{y+1}{y}=\frac{z+1}{z}$$
Find all superparticular positive integer triples.
Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.
A forest grows up $p$ percent during a summer, but gets reduced by $x$ units between two summers. At the beginning of this summer, the size of the forest has been $a$ units. How large should $x$ be if we want the forest to increase $q$ times in $n$ years?
We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$, where $a > 0$, there exists a transformation $B, B(x) = x + b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$, there is $y \in S$ such that $A(x) = B(y)$, and for any $t \in S$, there is a $u \in S$ such that $B(t) = A(u).$ Determine all superinvariant sets.
Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$
[color = red]The official statement does not have the final period.[/color]
If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.
For a word $w$ consisting of $n$ lowercase letters, an [i]edit[/i] is specified by a pair $(i,c)$ where $i\in \{1,\ldots, n\}$ and $c$ is a lowercase letter, and transforms $w$ by replacing its $i^{\text{th}}$ letter with $c$. It is possible that $c$ is the same as the letter it replaced.
How many sequences of six edits transform $\verb|frog|$ into $\verb|goat|$? Note that on the word $\verb|abcd|$, the edits $(1,\verb|a|)$ and $(2,\verb|b|)$ are considered distinct, even though both result in the word $\verb|abcd|$.
Given a positive integer $\displaystyle n = \prod_{i=1}^s p_i^{\alpha_i}$, we write $\Omega(n)$ for the total number $\displaystyle \sum_{i=1}^s \alpha_i$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n) = (-1)^{\Omega(n)}$ (so, for example, $\lambda(12)=\lambda(2^2\cdot3^1)=(-1)^{2+1}=-1$).
Prove the following two claims:
i) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = +1$;
ii) There are infinitely many positive integers $n$ such that $\lambda(n) = \lambda(n+1) = -1$.
[i](Romania) Dan Schwarz[/i]
At what $a$ does the graph of the function $y = x^4+x^3+ax$ have an axis of symmetry parallel to the axis $Oy$?
Let $AL_a$, $BL_b$, $CL_c$ be the bisecors of triangle $ABC$. The tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at point $K_a$, points $K_b$, $K_c$ are defined similarly. Prove that the lines $K_aL_a$, $K_bL_b$ and $K_cL_c$ concur.
Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$. Prove that
$$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$
Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$.
Hereby, $\mathbb{R}_+$ is the set of all positive real numbers.
[i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$.
A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.
Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$.
Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$?
If yes, find the smallest $k$. If not, prove this.
After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam?
[i]Proposed by Eric Wang[/i]
2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.