Found problems: 85335
If $x,y,z > 0, k>2$ and $a=x+ky+kz, b=kx+y+kz, c=kx+ky+z$, show that $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} \ge \frac{3}{2k+1}$.
The lengths of sides $CB$ and $CA$ of $\vartriangle ABC$ are $a$ and $b$, and the angle between them is $\gamma = 120^o$. Express the length of the bisector of $\gamma$ in terms of $a$ and $b$.
Let ABCD be a square and K
be a circle with diameter AB. For an
arbitrary point P on side CD, segments AP and BP meet K
again at
points M and N, respectively, and lines DM and CN meet at point Q.
Prove that Q lies on the circle K
and that AQ : QB = DP : PC.
On the line $g$ we are given the segment $AB$ and a point $C$ not on $AB$. Prove that on $g$, there exists at least one pair of points $P,Q$ symmetrical with respect to $C$, which divide the segment $AB$ internally and externally in the same ratios, i.e
$$\frac{PA}{PB}=\frac{QA}{QB}\qquad(1)$$
If $A,B,P,Q$ are such points from the line $g$ satisfying $(1)$, prove that the midpoint $C$ of the segment $PQ$ is the external point for the segment $AB$.
[i]K. Petrov[/i]
Suppose that $ a,b,c,p$ are real numbers with $ a,b,c$ not all equal, such that: $ a\plus{}\frac{1}{b}\equal{}b\plus{}\frac{1}{c}\equal{}c\plus{}\frac{1}{a}\equal{}p.$ Determine all possible values of $ p$ and prove that $ abc\plus{}p\equal{}0$.
Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.
There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant can cross its own path as it walks?
[i]Proposed by Gabriel Wu[/i]
The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.
Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t.
\[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]
Let $\mathbb{N}$ be the set of positive integers. Across all function $f:\mathbb{N}\to\mathbb{N}$ such that $$mn+1\text{ divides } f(m)f(n)+1$$ for all positive integers $m$ and $n$, determine all possible values of $f(101).$
Does there exist a polynomial $P(x)$ with integer coefficients such that $P(1+\sqrt[3]{2})=1+\sqrt[3]{2}$ and $P(1+\sqrt5)=2+3\sqrt5$?
For a prime number $p$ and a positive integer $n$, denote by $f(p, n)$ the largest integer $k$ such that $p^k \mid n!$. Let $p$ be a given prime number and let $m$ and $c$ be given positive integers. Prove that there exist infinitely many positive integers $n$ such that $f(p, n) \equiv c \pmod m$.
Find the largest positive integer that cannot be written in the form $a + bc$ for some positive integers $a, b, c$, satisfying $a < b < c$.
Felix chooses a positive integer as the starting number and writes it on the board. He then repeats the next step: he replaces the number $n$ on the board by $\frac12n$ if $n$ is even and by $n^2 + 3$ if $n$ is odd. For how many choices of starting numbers below $2023$ will Felix never write a number of more than four digits on the board?
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
Consider a parallelogram $ABCD$. $E$ is a point on ray $\overrightarrow{AD}$. $BE$ intersects $AC$ at $F$ and $CD$ at $G$. If $BF = EG$ and $BC = 3$, find the length of $AE$
In one peculiar family, the mother and the three children have exactly the same birthday. Currently, the mother is $37$ years old while each of children are $9$ years old. How old will the mother be when the sum of the ages of the three children equals her age?
$\textbf{(A) }14\qquad\textbf{(B) }27\qquad\textbf{(C) }42\qquad\textbf{(D) }57\qquad\textbf{(E) }66$
The $A$-excircle of the triangle $ABC$ touches the side $BC$ at point $K$. The circumcircles of triangles $AKB$ and $AKC$ intersect for the second time with the bisector of angle $A$ at points $X$ and $Y$ respectively. Let $M$ be the midpoint of $BC$. Prove that the circumcenter of triangle $XYM$ lies on $BC$.
Let G be a domain (connected open set) in the $R^2$ plane whose boundary is locally connected. Prove that for every point q of the boundary of G there exists a simple arc $v_q$ in which $q\in v_q$ and $v_q\setminus\{q\}\subset G$.
other questions:
(i) Show that local connectedness cannot be replaced by connectedness.
(ii) Show that if we replace the $R^2$ plane with $R^3$ space, the statement does not hold.
Let be the regular hexagonal prism $ABCDEFA'B C'D'E'F'$ with the base edge of $12$ and the height of $12 \sqrt{3}$. We denote by $N$ the middle of the edge $CC'$.
a) Prove that the lines $BF'$ and $ND$ are perpendicular
b) Calculate the distance between the lines $BF'$ and $ND$.
In a triangle $ABC$ with right angle $C$, the altitude $CH$ is drawn. An arbitrary circle passing through points $C$ and $H$ meets the segments $AC$, $CB$ and $BH$ for the second time at points $Q$, $P$ and $R$ respectively. Segments $HP$ and $CR$ meet at point $T$. What is greater: the area of triangle $CPT$ or the sum of areas of triangles $CQH$ and $HTR$? (5 marks)
a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then
$$A^n=\operatorname{tr}(A)A^{n-1}.$$
b) Does the assertion from a) also hold for odd positive integers $k$?
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that
\[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\]
is divisible by the product $c_1c_2\ldots c_n$.
Triangle $ABC$ has incircle $\omega$ and $A$-excircle $\omega_A.$ Circle $\gamma_B$ passes through $B$ and is externally tangent to $\omega$ and $\omega_A.$ Circle $\gamma_C$ passes through $C$ and is externally tangent to $\omega$ and $\omega_A.$ If $\gamma_B$ intersects line $BC$ again at $D,$ and $\gamma_C$ intersects line $BC$ again at $E,$ prove that $BD=EC.$
Equilateral triangle $ABC$ is inscribed in a circle with radius $6$. Find the area of the region enclosed by $AB$, $AC$, and the minor arc $BC$.