Found problems: 85335
Given a concyclic quadrilateral $ABCD$ with circumcenter $O$. Let $E$ be the intersection of $AD$ and $BC$, while $F$ be the intersection of $AC$ and $BD$. A circle $w$ are tangent to $BD$ and $AC$ such that $F$ is the orthocenter of $\triangle QEP$ where $PQ$ is a diameter of $w$. Prove that $EO$ passes through the center of $w$.
Let $n$ be an integer greater than $3$. Show that it is possible to divide a square into $n^2 + 1$ or more disjointed rectangles and with sides parallel to those of the square so that any line parallel to one of the sides intersects at most the interior of $n$ rectangles.
Note: We say that two rectangles are [i]disjointed [/i] if they do not intersect or only intersect at their perimeters.
Let $P$ and $Q$ be points on a circle $k$. A chord $AC$ of $k$ passes through the midpoint $M$ of $PQ$. Consider a trapezoid $ABCD$ inscribed in $k$ with $AB \parallel PQ \parallel CD$. Prove that the intersection point $X$ of $AD$ and $BC$ depends only on $k$ and $P,Q.$
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
Let $AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF,$ respectively, such that $BD$ is parallel to $EF.$ Point $C$ is chosen in the interior of triangle $AEF$ such that $ABCD$ is cyclic. If $BC = 3$ and $CD = 4,$ then the ratio of areas $\tfrac{[ABCD]}{[AEF]}$ can be written as $\tfrac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$.
Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.
Find all triples of nonnegative real numbers \((x, y, z)\) that satisfy the equality:
\[
\frac{\left(x^2 - y\right)(1 - y)}{(x - y)^2} + \frac{\left(y^2 - z\right)(1 - z)}{(y - z)^2} + \frac{\left(z^2 - x\right)(1 - x)}{(z - x)^2} = 3
\]
[i]Proposed by Vadym Solomka[/i]
The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event?
[i]Alexey Zaslavsky[/i]
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.
[i]Proposed by Iman Maghsoudi[/i]
Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.
Poland
Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.
Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$.
Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$. Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$. Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively.
Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$.
[i]Proposed by Andrew Gu and Frank Han[/i]
Find all four-digit natural numbers $\overline{xyzw}$ with the property that their sum plus the sum of their digits equals $2003$.
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
Let $S$ be the set of integers from $1$ to $13$ inclusive. A permutation of $S$ is a function $f : S \to S$ such that $f(x) \ne f(y)$ if $x \ne y$. For how many distinct permutations $f$ does there exists an $n $ such that $f^n(i) = 13 - i + 1$ for all $i$.
Let's consider all possible quadratic trinomials of the form $x^2 + ax + b$, where $a$ and $b$ are positive integers not exceeding some positive integer $N$. Prove that the number of pairs of such trinomials having a common root does not exceed $N^2$.
Does there exists a subset of positive integers with infinite members such that for every two members $a,b$ of this set
\[a^2-ab+b^2|(ab)^2\]
In convex hexagonal pyramid 11 edges are equal to 1.Find all possible values of 12th edge.
Evaluate: $ \sum\limits_{n\equal{}1}^\infty \arctan{\left(\frac{1}{n^2\minus{}n\plus{}1}\right)}$
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.
A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$
What is the probability of having at most fi ve diff erent digits in the sequence?
For any number $x$, let $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. A sequence $a_1,a_2,\cdots$ is given, where \[a_n=\left\lfloor{\sqrt{2n}+\dfrac{1}{2}}\right\rfloor.\]
How many values of $k$ are there such that $a_k=2010$?
Let $ a,b,c$ be real numbers such that $ a\plus{}b\plus{}c\equal{}3$. Prove that
\[\frac{1}{5a^2\minus{}4a\plus{}11}\plus{}\frac{1}{5b^2\minus{}4b\plus{}11}\plus{}\frac{1}{5c^2\minus{}4c\plus{}11}\leq\frac{1}{4}\]
$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:
- there is only one winner;
- there are $\displaystyle 3$ students on the second place;
- no student lost all $\displaystyle 4$ matches.
How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)
Let $x=\frac pq$ for $p$, $q$ coprime. Find $p+q$.
[asy]
import olympiad;
size(200);
pen qq=font("phvb");
defaultpen(linewidth(0.6)+fontsize(10pt));
pair A=(-2.25,7),B=(-5,0),C=(5,0),D=waypoint(A--B,3/7), E=waypoint(A--C,1/2),F=intersectionpoint(C--D, B--E);
draw(A--B--C--cycle^^B--E^^C--D);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NW);
label("$E$",E,NE);
label("$F$",F,N);
label(scale(2.5)*"X",centroid(A,D,E),qq);
label(scale(2.5)*"3",centroid(B,D,F),0.5*N,qq);
label(scale(2.5)*"6",centroid(B,F,C),0.25*dir(180),qq);
label(scale(2.5)*"2",centroid(C,E,F),dir(140),qq);
[/asy]