Found problems: 85335
Determine all pairs of $(m, n)$ such that is possible to tile the table $ m \times n$ with figure ”corner” as in figure with condition that in that tilling does not exist rectangle (except $m \times n$) regularly covered with figures.
Determine the lowest positive integer n such that following statement is true:
If polynomial with integer coefficients gets value 2 for n different integers,
then it can't take value 4 for any integer.
A circle is drawn on the plane. How to use only a ruler to draw a perpendicular from a given point outside the circle to a given line passing through the center of this circle?
Let $ABC$ be an acute triangle with orthocenter $H$. Let $D$ and $E$ be feet of the altitudes from $B$ and $C$ respectively. Let $M$ be the midpoint of segment $AH$ and $F$ be the intersection point of $AH$ and $DE$. Furthermore, let $P$ and $Q$ be the points
inside triangle $ADE$ so that $P$ is an intersection of $CM$ and the circumcircle of $DFH$, and $Q$ is an intersection of $BM$ and the circumcircle of $EFH$. Prove that the intersection of lines $DQ$ and $EP$ lies on segment $AH$.
Let $A$ and $B$ be points on the circle $k$ with center $O$, so that $AB> AO$. Let $C$ be the intersection of the bisectors of $\angle OAB$ and $k$, different from $A$. Let $D$ be the intersection of the straight line $AB$ with the circumcircle of the triangle $OBC$, different from $B$. Show that $AD = AO$ .
A word of length $n$ that consists only of the digits $0$ and $1$, is called a bit-string of length $n$. (For example, $000$ and $01101$ are bit-strings of length 3 and 5.) Consider the sequence $s(1), s(2), ...$ of bit-strings of length $n > 1$ which is obtained as follows :
(1) $s(1)$ is the bit-string $00...01$, consisting of $n - 1$ zeros and a $1$ ;
(2) $s(k+1)$ is obtained as follows :
(a) Remove the digit on the left of $s(k)$. This gives a bit-string $t$ of length $n - 1$.
(b) Examine whether the bit-string $t1$ (length $n$, adding a $1$ after $t$) is already in $\{s(1), s(2), ..., s(k)\}$. If this is the not case, then $s(k+1) = t1$. If this is the case then $s(k+1) = t0$.
For example, if $n = 3$ we get :
$s(1) = 001 \rightarrow s(2) = 011 \rightarrow s(3) = 111 \rightarrow s(4) = 110 \rightarrow s(5) = 101$
$\rightarrow s(6) = 010 \rightarrow s(7) = 100 \rightarrow s(8) = 000 \rightarrow s(9) = 001 \rightarrow ...$
Suppose $N = 2^n$.
Prove that the bit-strings $s(1), s(2), ..., s(N)$ of length $n$ are all different.
Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.
Prove that there are two numbers $u$ and $v$, between any $101$ real numbers that apply $100 |u - v| \cdot |1 - uv| \le (1 + u^2)(1 + v^2)$
Snow White has, in her huge basket, $2021$ apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous.
To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die.
(a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days.
(b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal?
b) Show that if the product of all the positive divisors of two natural numbers are equal, then the two numbers must be equal.
Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that
(1) All elements of $A_n$ are rational.
(2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.
Let $x, y$, and $z$ be three not necessarily real numbers that satisfy the following system of equations:
$x^3 -4 = (2y +1)^2$
$y^3 -4 = (2z +1)^2$
$z^3 -4 = (2x +1)^2$.
Find the greatest possible real value of $(x -1)(y -1)(z -1)$.
Let $ABC$ be a right triangle with $m(\widehat {C}) = 90^\circ$, and $D$ be its incenter. Let $N$ be the intersection of the line $AD$ and the side $CB$. If $|CA|+|AD|=|CB|$, and $|CN|=2$, then what is $|NB|$?
Find all functions $f : R \to R$ satisfying $f(x + xy + f(y))=(f(x) + \frac12)(f(y) + \frac12 )$ for all $x, y \in R$.
(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also
\[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\]
(b) Show that there are no two positive integers $a$ and $b$ such that
\[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $D$ be the point inside triangle $ABC$ with the property that $\overline{BD} \perp \overline{CD}$ and $\overline{AD} \perp \overline{BC}$. Then the length $AD$ can be expressed in the form $m-\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by Michael Ren[/i]
In the figure above, the large triangle and all four shaded triangles are equilateral. If the areas of triangles $A, B,$ and $C$ are $1, 2,$ and $3,$ respectively, compute the smallest possible integer ratio between the area of the entire triangle to the area of triangle $D.$
[Insert Diagram]
[i]Proposed by Alex Li[/i]
An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.
A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right.
All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.
A rectangular board of 8 columns has squared numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips 3 squares and shades square 10, and continues in this way until there is at least one shaded square in each column. What is the number of the shaded square that first achieves this result?
[asy]
unitsize(20);
for(int a = 0; a < 10; ++a)
{
draw((0,a)--(8,a));
}
for (int b = 0; b < 9; ++b)
{
draw((b,0)--(b,9));
}
draw((0,0)--(0,-.5));
draw((1,0)--(1,-1.5));
draw((.5,-1)--(1.5,-1));
draw((2,0)--(2,-.5));
draw((4,0)--(4,-.5));
draw((5,0)--(5,-1.5));
draw((4.5,-1)--(5.5,-1));
draw((6,0)--(6,-.5));
draw((8,0)--(8,-.5));
fill((0,8)--(1,8)--(1,9)--(0,9)--cycle,black);
fill((2,8)--(3,8)--(3,9)--(2,9)--cycle,black);
fill((5,8)--(6,8)--(6,9)--(5,9)--cycle,black);
fill((1,7)--(2,7)--(2,8)--(1,8)--cycle,black);
fill((6,7)--(7,7)--(7,8)--(6,8)--cycle,black);
label("$2$",(1.5,8.2),N);
label("$4$",(3.5,8.2),N);
label("$5$",(4.5,8.2),N);
label("$7$",(6.5,8.2),N);
label("$8$",(7.5,8.2),N);
label("$9$",(0.5,7.2),N);
label("$11$",(2.5,7.2),N);
label("$12$",(3.5,7.2),N);
label("$13$",(4.5,7.2),N);
label("$14$",(5.5,7.2),N);
label("$16$",(7.5,7.2),N);
[/asy]
$\text{(A)}\ 36 \qquad \text{(B)}\ 64 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 120$
Given a convex pentagon $ABCDE$, with $\angle BAC = \angle ABE = \angle DEA - 90^o$, $\angle BCA = \angle ADE$ and also $BC = ED$. Prove that $BCDE$ is parallelogram.
Let $a, b,$ and $c$ be positive integers such that $a+b+c=23$ and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^{2}+b^{2}+c^{2}$?
$\textbf{(A)} ~259\qquad\textbf{(B)} ~438\qquad\textbf{(C)} ~516\qquad\textbf{(D)} ~625\qquad\textbf{(E)} ~687$
Proposed by [b]djmathman[/b]
Let $K$ be a point inside an acute triangle $ ABC $, such that $ BC $ is a common tangent of the circumcircles of $ AKB $ and $ AKC$. Let $ D $ be the intersection of the lines $ CK $ and $ AB $, and let $ E $ be the intersection of the lines $ BK $ and $ AC $ . Let $ F $ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $ F$ and radius $FD$ intersect at points $P$ and $Q$.
Prove that the segment $PQ$ is a diameter of $k$.
For all positive integers $n,$ let $r_n$ be defined as \[r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.\]Prove that $\sum_{r=1}^\infty r_i=0.$
Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$. Prove that $CNMF$ is a cyclic quadrilateral.