Found problems: 85335
[list=a]
[*]Determine the positive integer $x$ for which $\dfrac14-\dfrac{1}{x}=\dfrac16.$
[*]Determine all pairs of positive integers $(a,b)$ for which $ab-b+a-1=4.$
[*]Determine the number of pairs of positive integers $(y,z)$ for which $\dfrac{1}{y}-\dfrac{1}{z}=\dfrac{1}{12}.$
[*]Prove that, for every prime number $p$, there are at least two pairs $(r,s)$ of positive integers for which $\dfrac{1}{r}-\dfrac{1}{s}=\dfrac{1}{p^2}.$[/list]
Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.
There exist positive integers $A,B$ and $C$, with no common factor greater than $1$, such that \[A \log_{200} 5 + B \log_{200} 2 = C. \] What is $A+B+C$?
$\textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7\qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$
Let $a>0$ and $12a+5b+2c>0$. Prove that it is impossible for the equation
$ax^2+bx+c=0$ to have two real roots in the interval $(2,3)$.
Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order).
$r$ is the tangent to $\Gamma$ at point A.
$s$ is the tangent to $\Gamma$ at point D.
Let $E=r \cap BC,F=s \cap BC$.
Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$
Show that the points $X,Y,Z$ are collinear.
Note: assume the existence of all above points.
Inside the quadrilateral $ABCD$ marked a point $O$ such that $\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o$. Prove that the centers of the circumscribed circles around triangles $OAD$ and $OBC$ as well as the midpoints of the sides $AB$ and $CD$ lie on one circle.
(Anton Trygub)
Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.
The value of $ \frac{1}{16}a^0\plus{}\left (\frac{1}{16a} \right )^0\minus{} \left (64^{\minus{}\frac{1}{2}} \right )\minus{} (\minus{}32)^{\minus{}\frac{4}{5}}$ is:
$ \textbf{(A)}\ 1 \frac{13}{16} \qquad
\textbf{(B)}\ 1 \frac{3}{16} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ \frac{7}{8} \qquad
\textbf{(E)}\ \frac{1}{16}$
Andriy and Olesya take turns (Andriy starts) in a $2 \times 1$ rectangle, drawing horizontal segments of length $2$ or vertical segments of length $1$, as shown in the figure below.
[img]https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png[/img]
After each move, the value $P$ is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move $P$ is divisible by $2021$ for the first time. Who has a winning strategy?
[i]Proposed by Bogdan Rublov[/i]
Let $ABC$ be a triangle with $AB<AC$, and let $I_a$ be its $A$-excenter. Let $D$ be the projection of $I_a$ to $BC$. Let $X$ be the intersection of $AI_a$ and $BC$, and let $Y,Z$ be the points on $AC,AB$, respectively, such that $X,Y,Z$ are on a line perpendicular to $AI_a$. Let the circumcircle of $AYZ$ intersect $AI_a$ again at $U$. Suppose that the tangent of the circumcircle of $ABC$ at $A$ intersects $BC$ at $T$, and the segment $TU$ intersects the circumcircle of $ABC$ at $V$. Show that $\angle BAV=\angle DAC$.
[i]Proposed by usjl.[/i]
Each of the players in a tennis tournament played one match against each of the others. If every player won at least one match, show that there are three players A,B, and C such that A beats B, B beats C, and C beats A. Such a triple of player is called a cycle. Determine the number of maximum cycles such a tournament can have.
Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.
Inside a square of sidelength $ 1$ there are finitely many disks that are allowed to overlap. The sum of all circumferences is $ 10$. Show that there is a line intersecting or touching at least $ 4$ disks.
A perpendicular bisector to the side $AC$ of triangle $ABC$ meets $BC,AB$ at points $A_1$ and $C_1$ respectively. Points $O,O_1$ are the circumcenters of triangles $ABC$ and $A_1BC_1$ respectively. Prove that $C_1O_1\perp AO$.
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
Inside a rectangular piece of paper $n$ rectangular holes with sides parallel to the sides of the paper have been cut out. Into what minimal number of rectangular pieces (without holes) is it always possible to cut this piece of paper?
(A Shapovalov)
In $ \triangle ABC$, we have $ \angle C \equal{} 3 \angle A$, $ a \equal{} 27$, and $ c \equal{} 48$. What is $ b$?
[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, B=(14,0), C=(10,6);
draw(A--B--C--cycle);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, N);
label("$a$", B--C, dir(B--C)*dir(-90));
label("$b$", A--C, dir(C--A)*dir(-90));
label("$c$", A--B, dir(A--B)*dir(-90));
[/asy]
$ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 37 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ \text{not uniquely determined}$
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?
$\textbf{(A) } 27 \qquad\textbf{(B) } 16\sqrt3 \qquad\textbf{(C) } 12+10\sqrt3 \qquad\textbf{(D) } 9+12\sqrt3 \qquad\textbf{(E) } 30$
Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
[i]Proposed by Andy Xu[/i]
Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.
Determine all functions $f:\mathbb{R}\to\mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers, satisfying the following two conditions:
1) There exists a real number $M$ such that for every real number $x,f(x)<M$ is satisfied.
2) For every pair of real numbers $x$ and $y$,
\[ f(xf(y))+yf(x)=xf(y)+f(xy)\]
is satisfied.
For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$.
Fi nd the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$?
[i]Based on a proposal by Neelabh Deka[/i]