Found problems: 85335
Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.
Find the number of counter examples to the statement:
\[``\text{If N is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then N is prime}."\]
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $
In rectangle $ABCD$, the points $M,N,P, Q$ lie on $AB$, $BC$, $CD$, $DA$ respectively such that the area of triangles $AQM$, $BMN$, $CNP$, $DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram.
by Mahdi Etesami Fard
For a positive integer $k$, denote by $f(k)$ the number of positive integer $m$ such that the remainder of $km$ modulo $2019^3$ is greater than $m$. Find the amount of different numbers among $f(1), f(2), ..., f(2019^3)$.
Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop."
[list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot?
[b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to
the empty spot?[/list]
Let $n\in\mathbb{N}$, the set $A=\{(x_1,x_2...,x_n)|x_i\in\mathbb{R}_{+}, i=1,2,...,n\}$ and the function $$f:A\rightarrow\mathbb{R}, f(x_1,...,x_n)=\frac{1}{x_1}+\frac{1}{2x_2}+\ldots+\frac{1}{(n-1)x_{n-1}}+\frac{1}{nx_n}.$$
Prove that $f(\textstyle\binom{n}{1},\binom{n}{2},...,\binom{n}{n-1},\binom{n}{n})=f(2^{n-1},2^{n-2},...,2,1).$
A number line with the integers $1$ through $20,$ from left to right, is drawn. Ten coins are placed along this number line, with one coin at each odd number on the line. A legal move consists of moving one coin from its current position to a position of strictly greater value on the number line that is not already occupied by another coin. How many ways can we perform two legal moves in sequence, starting from the initial position of the coins (different two-move sequences that result in the same position are considered distinct)?
A set of $n$ distinct integer weights $w_1,w_2,\ldots, w_n$ is said to be [i]balanced[/i] if after removing any one of weights, the remaining $(n-1)$ weights can be split into two subcollections (not necessarily with equal size)with equal sum.
$a)$ Prove that if there exist [i]balanced[/i] sets of sizes $k,j$ then also a [i]balanced[/i] set of size $k+j-1$.
$b)$ Prove that for all [i]odd[/i] $n\geq 7$ there exist a [i]balanced[/i] set of size $n$.
Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$. The area of the "bat wings" is
[asy]
size(180);
defaultpen(fontsize(11pt));
draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0));
draw((3,0)--(1,4)--(0,0));
fill((0,0)--(1,4)--(1.5,3)--cycle, black);
fill((3,0)--(2,4)--(1.5,3)--cycle, black);
label("$D$",(0,4),NW);
label("$C$",(1,4),N);
label("$B$",(2,4),N);
label("$A$",(3,4),NE);
label("$E$",(0,0),SW);
label("$F$",(3,0),SE);[/asy]
$\textbf{(A) }2\qquad\textbf{(B) }2 \frac{1}{2}\qquad\textbf{(C) }3\qquad\textbf{(D) }3 \frac{1}{2}\qquad \textbf{(E) }5$
Determine the smallest natural number $n$ such that $n^n$ is not a divisor of the product $1\cdot 2\cdot 3\cdot ... \cdot 2015\cdot 2016$.
Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.
$6$ pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni?
$ \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78 $
Find $a+b+c$, where $a,b,$ and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$.
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
Prove that among all triangles with inradius $1$, the equilateral one has the smallest perimeter .
Consider an infinite white plane divided into square cells. For which $k$ it is possible to paint a positive finite number of cells black so that on each horizontal, vertical and diagonal line of cells there is either exactly $k$ black cells or none at all?
A. Dinev, K. Garov, N Belukhov
A classroom has $30$ seats arranged into $5$ rows of $6$ seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?
Let $n \ge 2$ be an integer and $f_1(x), f_2(x), \ldots, f_{n}(x)$ a sequence of polynomials with integer coefficients. One is allowed to make moves $M_1, M_2, \ldots $ as follows: in the $k$-th move $M_k$ one chooses an element $f(x)$ of the sequence with degree of $f$ at least $2$ and replaces it with $(f(x) - f(k))/(x-k)$. The process stops when all the elements of the sequence are of degree $1$. If $f_1(x) = f_2(x) = \cdots = f_n(x) = x^n + 1$, determine whether or not it is possible to make appropriate moves such that the process stops with a sequence of $n$ identical polynomials of degree 1.
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$,
for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.
In each of $n$ houses on a straight street are one or more boys. At what point should all the boys meet so that the sum of the distances that they walk is as small as possible?
Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl."
"Abby is between Bret and Carl."
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
$\text{(A)}\ \text{Abby} \qquad \text{(B)}\ \text{Bret} \qquad \text{(C)}\ \text{Carl} \qquad \text{(D)}\ \text{Dana} \qquad \text{(E)}\ \text{There is not enough information to be sure.}$
In the following sum: $1 + 2 + 3 + 4 + 5 + 6$, if we remove the first two “+” signs, we obtain the new sum $123 + 4 + 5 + 6 = 138$. By removing three “$+$” signs, we can obtain $1 + 23 + 456 = 480$. Let us now consider the sum $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$, in which some “$+$” signs are to be removed. What are the three smallest multiples of $100$ that we can get in this way?
All integer numbers are colored in 3 colors in arbitrary way. Prove that there are two distinct numbers whose difference is a perfect square and the numbers are colored in the same color.
Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions
\begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\
g'&=fg^2h+\frac4{fh},\ g(0)=1,\\
h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*}
Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$