Found problems: 85335
Find all four-digit perfect squares such that:
$\bullet$ All your figures are less than $9$.
$\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.
For every natural $n$ let $a_n=20\dots 018$ with $n$ ceros, for example, $a_1=2018, a_3=200018, a_7=2000000018$. Prove that there are infinity values of $n$ such that $2018$ divides $a_n$
Let $A, B, C, D$ be four distinct points in the plane such that the length of the six line segments $AB, AC, AD, BC, BD, CD$ form a $2$-element set ${a, b}$. If $a > b$, determine all the possible values of $\frac ab$.
A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.
A bored student walks down a hallway where there is a row of closed lockers, numbered from $1$ to $1024$. Opens cabinet No. $1$, then skips one cabinet and opens the next, and so on successively. When he reaches the end of the row, he turns around and starts again: he opens the first cabinet it finds closed, he skips the next closed cabinet and so on until the start from the hallway. goes from beginning to end, from end to beginning of the corridor until all the cabinets are left open. What is the number of the last cabinet he opened?
Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$
Prove that $ \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , $ for nonnegative $ m,n. $
[i]Gheorghe Stoica[/i]
Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric?
[i]Proposed by Gabriel Wu[/i]
Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular.
(B. Kukushkin)
Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.
Suppose that $ a,b$ are positive integers for which $ A\equal{}\frac{a\plus{}1}{b}\plus{}\frac{b}{a}$ is an integer.Prove that $ A\equal{}3$.
Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circumcircle of $\vartriangle XY Z$ at the point $W\ne X$. If the ratio $\frac{ WY}{WZ}$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.
There are $n$ boys $a_1, a_2, \ldots, a_n$ and $n$ girls $b_1, b_2, \ldots, b_n $. Some pairs of them are connected. Any two boys or two girls are not connected, and $a_i$ and $b_i$ are not connected for all $i \in \{ 1,2,\ldots,n\}$. Now all boys and girls are divided into several groups satisfying two conditions:
(i) Every groups contains an equal number of boys and girls.
(ii) There is no connected pair in the same group.
Assume that the number of connected pairs is $m$. Show that we can make the number of groups not larger than $\max\left \{2, \dfrac{2m}{n} +1\right \}$.
In a school there are $40$ different clubs, each of them contains exactly $30$ children. For every $i$ from $1$ to $30$ define $n_i$ as a number of children who attend exactly $i$ clubs. Prove that it is possible to organize $40$ new clubs with $30$ children in each of them such, that the analogical numbers $n_1, n_2,..., n_{30}$ will be the same for them.
Let $BD$ be a bisector of triangle $ABC$. Points $I_a$, $I_c$ are the incenters of triangles $ABD$, $CBD$ respectively. The line $I_aI_c$ meets $AC$ in point $Q$. Prove that $\angle DBQ = 90^\circ$.
A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. How many square tiles are there in the box?
$ \textbf{(A) }3 \qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }\text{11} $
At the beginning of a month a shop has $10$ different products for sale, each with equal prices. Every day the price of each product is either doubled or trebled. By the beginning of the following month all the prices have become different. Prove that the ratio (the maximal price) /(the minimal price) is greater than $27$.
(D. Fomin and Stanislav Smirnov, St Petersburg)
On an $n$ × $n$ board, the set of all squares that are located on or below the main diagonal of the board is called the$n-ladder$. For example, the following figure shows a $3-ladder$:
[asy]
draw((0,0)--(0,3));
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,0)--(1,3));
draw((0,2)--(2,2));
draw((2,0)--(2,2));
draw((0,3)--(1,3));
draw((3,0)--(3,1));
[/asy]
In how many ways can a $99-ladder$ be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?
Find the remainder of
$$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$
when divided by $101$.
How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?
$\textbf{(A) }6 \qquad\textbf{(B) }12 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 24 \qquad \textbf{(E) } 36$
[asy] import three;
currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */
draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle);
draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1));
draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1));
draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle);
label("$D$",(0,0,0),S);
label("$A$",(0,0,1),N);
label("$H$",(0,1,0),S);
label("$E$",(0,1,1),N);
label("$C$",(1,0,0),S);
label("$B$",(1,0,1),N);
label("$G$",(1,1,0),S);
label("$F$",(1,1,1),N);
[/asy]
Determine all positive integers $n$ such that $\frac{a^2+n^2}{b^2-n^2}$ is a positive integer for some $a,b\in \mathbb{N}$.
$Turkey$
Arnav is placing three rectangles into a $3 \times 3$ grid of unit squares. He has a $1\times 3$ rectangle, a $1\times 2$ rectangle, and a $1\times 1$ rectangle. He must place the rectangles onto the grid such that the edges of the rectangles align with the gridlines of the grid. If he is allowed to rotate the rectangles, how many ways can he place the three rectangles into the grid, without overlap?
[i]Proposed by William Yue[/i]
Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$.
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
The negation of the statement "all men are honest," is:
$ \textbf{(A)}\ \text{no men are honest} \qquad\textbf{(B)}\ \text{all men are dishonest} \\
\textbf{(C)}\ \text{some men are dishonest} \qquad\textbf{(D)}\ \text{no men are dishonest} \\
\textbf{(E)}\ \text{some men are honest}$