Found problems: 85335
Let A,B,C be three collinear points and a circle T(A,r).
If M and N are two diametrical opposite variable points on T,
Find locus geometrical of the intersection BM and CN.
Find all number sets $(a,b,c,d)$ s.t. $1 < a \le b \le c \le d$, $a,b,c,d \in \mathbb{N}$, and $a^2+b+c+d$, $a+b^2+c+d$, $a+b+c^2+d$, and $a+b+c+d^2$ are all square numbers. Sum the value of $d$ across all solution set(s).
Nils has a telephone number with eight different digits.
He has made $28$ cards with statements of the type “The digit $a$ occurs earlier than the digit $b$ in my telephone number” – one for each pair of digits appearing in his number.
How many cards can Nils show you without revealing his number?
Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer is $S$ is [i]also[/i] removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?
$\textbf{(A)} ~36.2 \qquad\textbf{(B)} ~36.4 \qquad\textbf{(C)} ~36.6 \qquad\textbf{(D)} ~36.8 \qquad\textbf{(E)} ~37$
Arrange the positive integers into two lines as follows:
\begin{align*} 1 \quad 3 \qquad 6 \qquad\qquad\quad 11 \qquad\qquad\qquad\qquad\quad\ 19\qquad\qquad32\qquad\qquad 53\ldots\\
\mbox{\ \ } 2 \quad 4\ \ 5 \quad 7\ \ 8\ \ 9\ \ 10\quad\ 12\ 13\ 14\ 15\ 16\ 17\ 18\quad\ 20 \mbox{ to } 31\quad\ 33 \mbox{ to } 52\quad\ \ldots\end{align*} We start with writing $1$ in the upper line, $2$ in the lower line and $3$ again in the upper line. Afterwards, we alternately write one single integer in the upper line and a block of integers in the lower line. The number of consecutive integers in a block is determined by the first number in the previous block.
Let $a_1$, $a_2$, $a_3$, $\ldots$ be the numbers in the upper line. Give an explicit formula for $a_n$.
Find all $3$-tuples of positive reals $(a,b,c)$ such that
$$\begin{cases}a\sqrt[2019]b-c=a\\b\sqrt[2019]c-a=b\\c\sqrt[2019]a-b=c\end{cases}$$
How many positive integer tuples $ (x_1,x_2,\dots, x_{13})$ are there satisfying the inequality $x_1+x_2+\dots + x_{13}\leq 2006$?
$
\textbf{(A)}\ \frac{2006!}{13!1993!}
\qquad\textbf{(B)}\ \frac{2006!}{14!1992!}
\qquad\textbf{(C)}\ \frac{1993!}{12!1981!}
\qquad\textbf{(D)}\ \frac{1993!}{13!1980!}
\qquad\textbf{(E)}\ \text{None of above}
$
Given two integers $n\geq 1$ and $q\geq 2$, let $A=\{(a_1,\ldots ,a_n):a_i\in\{0,\ldots ,q-1\}, i=1,\ldots ,n\}$. If $a=(a_1,\ldots ,a_n)$ and $b=(b_1,\ldots ,b_n)$ are two elements of $A$, let $\delta(a,b)=\#\{i:a_i\neq b_i\}$. Let further $t$ be a non-negative integer and $B$ a non-empty subset of $A$ such that $\delta(a,b)\geq 2t+1$, whenever $a$ and $b$ are distinct elements of $B$. Prove that the two statements below are equivalent:
a) For any $a\in A$, there is a unique $b\in B$, such that $\delta (a,b)\leq t$;
b) $\displaystyle|B|\cdot \sum_{k=0}^t \binom{n}{k}(q-1)^k=q^n$
Two players take turns entering a symbol in an empty cell of a $1 \times n$ chessboard, where $n$ is an integer greater than $1$. Aaron always enters the symbol $X$ and Betty always enters the symbol $O$. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which player has a winning strategy?
Let $a_1,\dots,a_n$ be a non increasing sequence of positive real numbers. Prove that \[\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}.\] When does equality hold?
Solve the equation $\big(2cos(x-\frac{\pi}{4})+tgx\big)^3=54 sin^2x$, $x\in \big[0,\frac{\pi}{2}\big)$
Let $\delta_0=\triangle A_0B_0C_0$ be a triangle. On each of the sides $B_0C_0$, $C_0A_0$, $A_0B_0$, there are constructed squares in the halfplane, not containing the respective vertex $A_0,B_0,C_0$ and $A_1,B_1,C_1$ are the centers of the constructed squares. If we use the triangle $\delta_1=\triangle A_1B_1C_1$ in the same way we may construct the triangle $\delta_2=\triangle A_2B_2C_2$; from $\delta_2=\triangle A_2B_2C_2$ we may construct $\delta_3=\triangle A_3B_3C_3$ and etc. Prove that:
(a) segments $A_0A_1,B_0B_1,C_0C_1$ are respectively equal and perpendicular to $B_1C_1,C_1A_1,A_1B_1$;
(b) vertices $A_1,B_1,C_1$ of the triangle $\delta_1$ lies respectively over the segments $A_0A_3,B_0B_3,C_0C_3$ (defined by the vertices of $\delta_0$ and $\delta_1$) and divide them in ratio $2:1$.
[i]K. Dochev[/i]
Kelvin the frog is on the bottom-left lily pad of a $3 \times 3$ grid of lily pads, and his home is at the top right lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove $4$ of the lily pads so that the bottom-left and top-right lily pads both remain, but Kelvin cannot get home.
Let $ A\equal{}\{1,2,\ldots, 2008\}$. We will say that set $ X$ is an $ r$-set if $ \emptyset \neq X \subset A$, and $ \sum_{x\in X} x \equiv r \pmod 3$. Let $ X_r$, $ r\in\{0,1,2\}$ be the set of $ r$-sets.
Find which one of $ X_r$ has the most elements.
A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player wins, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?
All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?
Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$, $| f (0) | \le 1$ and $| f (1) | \le 1$. Show that if $-1 \le x \le 1$, then $| f (x) | \le \frac54$ and $| g (x) | \le 2$.
A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0.
a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared".
b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously :
1) $e-2$ , $e$ , $e+4$ are all prime numbers.
2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.
Determine all positive integers $n{}$ for which there exist pairwise distinct integers $a_1,\ldots,a_n{}$ and $b_1,\ldots, b_n$ such that \[\prod_{i=1}^n(a_k^2+a_ia_k+b_i)=\prod_{i=1}^n(b_k^2+a_ib_k+b_i)=0, \quad \forall k=1,\ldots,n.\]
Let $ ABC $ be an isosceles triangle with $ AC=BC $ . Take points $ D $ on side $AC$ and $E$ on side $BC$ and $ F $ the intersection of bisectors of angles $ DEB $ and $ADE$ such that $ F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.
Given a $2\times 5$ rectangle is divided into unit squares as figure below.
[img]https://cdn.artofproblemsolving.com/attachments/6/a/9432bbf40f6d89ee1cbb507e1a3f65326c6a13.png[/img]
How many ways are there to write the letters $H, A, N, O, I$ into all of the unit squares, such that two neighbor squares (the squares with a common side) do not contain the same letters? (Each unit square is filled by only one letter and each letter may be used several times or not used as well.)
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\left\lceil \sqrt{s} \right\rceil \mid s \in S\}$. Find the number of sets $T$ such that $\vert f(T) \vert = 2$ and $f(f(T)) = \{2\}$.
We consider the following game for one person. The aim of the player is to reach a fixed capital $C>2$. The player begins with capital $0<x_0<C$. In each turn let $x$ be the player’s current capital. Define $s(x)$ as follows:
$$s(x):=\begin{cases}x&\text{if }x<1\\C-x&\text{if }C-x<1\\1&\text{otherwise.}\end{cases}$$Then a fair coin is tossed and the player’s capital either increases or decreases by $s(x)$, each with probability $\frac12$. Find the probability that in a finite number of turns the player wins by reaching the capital $C$.
Given an integer $n$. We rearrange the digits of $n$ to get another number $m$. Prove that it is impossible to get $m+n = 999999999$.
Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle BCD = \theta$ for some acute angle $\theta$. Point $X$ lies inside the quadrilateral such that $\angle XAD = \angle XDA = 90^{\circ}-\theta$. Prove that $BX = XC$.