Found problems: 85335
Let $ABC$ be an acute-angled triangle with the property that the bisector of $\angle BAC$, the altitude through $B$ and the perpendicular bisector of $AB$ intersect in one point. Determine the angle $\alpha = \angle BAC$.
Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality
\[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\]
holds. Determine all cases of equality.
[i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]
Find the smallest prime number that does not divide \[9+9^2+9^3+\cdots+9^{2010}.\]
$a_1,a_2,\ldots$ is a sequence of [u]nonzero integer[/u] numbers that for all $n\in\mathbb{N}$, if $a_n=2^\alpha k$ such that $k$ is an odd integer and $\alpha$ is a nonnegative integer then: $a_{n+1}=2^\alpha-k$. Prove that if this sequence is periodic, then for all $n\in\mathbb{N}$ we have: $a_{n+2}=a_n$. (The sequence $a_1,a_2,\ldots$ is periodic iff there exists natural number $d$ that for all $n\in\mathbb{N}$ we have: $a_{n+d}=a_n$)
Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively.
(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.
(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.
Let $ABCDEF$ be a hexagon with $AB||DE,\ BC||EF,\ CD||FA$ and in which the diagonals $AD,BE$ and $CF$ are congruent. Prove that the hexagon can be inscribed in a circle.
You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\
bc + bd = 5c + 5d \\
ac + cd = 7a + 7d \\
ad + bd = 9a + 9b \end{cases} $
Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$
Let $x,y,z \in [0,1]$ , and $|y-z|\leq \frac{1}{2},|z-x|\leq \frac{1}{2},|x-y|\leq \frac{1}{2}$ . Find the maximum and minimum value of $W=x+y+z-yz-zx-xy$.
For any positive integer $n$, define the subset $S_n$ of natural numbers as follow
$$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$
Find all positive integers $n$ such that there exists an element of $S_n$ which [u]doesn't belong[/u] to any of the sets $S_1, S_2,\dots,S_{n-1}$.
[i]Proposed by Yahya Motevassel[/i]
Consider two odd natural numbers $a$ and $b$ where $a$ is a divisor of $b^2+2$ and $b$ is a divisor of $a^2+2.$ Prove that $a$ and $b$ are the terms of the series of natural numbers $\langle v_n\rangle$ defined by
\[v_1 = v_2 = 1; v_n = 4v_ {n-1}-v_{n-2} \ \ \text{for} \ n\geq 3.\]
Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\tfrac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn?
$ \textbf{(A)}\ 0.75 \qquad
\textbf{(B)}\ 0.8 \qquad
\textbf{(C)}\ 1.35 \qquad
\textbf{(D)}\ 1.5 \qquad
\textbf{(E)}\ 3$
The sequence $\{x_n\}$ of real numbers is defined by
\[x_1=1 \quad\text{and}\quad x_{n+1}=x_n+\sqrt[3]{x_n} \quad\text{for}\quad n\geq 1.\]
Show that there exist real numbers $a, b$ such that $\lim_{n \rightarrow \infty}\frac{x_n}{an^b} = 1$.
Show that the equation $ y^{37} \equal{} x^3 \plus{}11\pmod p$ is solvable for every prime $ p$, where $ p\le 100$.
In triangle $\triangle ABC$ , $M, N$ are midpoints of $AC,AB$ respectively. Assume that $BM,CN$ cuts $(ABC)$ at $M',N'$ respectively. Let $X$ be on the extention of $BC$ from $B$ st $\angle N'XB=\angle ACN$. And define $Y$ similarly on the extention of $BC$ from $C$. Prove that $AX=AY$.
Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$.
Prove that $\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}$
In a $100$ by $100$ grid, there is a spider and $100$ bugs. Each time, the spider can walk up, down, left or right, and the spider aims to visit all the squares with bugs to eat them all. The spider begins from the top-left corner. Show that no matter where the bugs are, the spider can always eat them all within $2000$ steps.
Given real $c > -2$. Prove that for positive reals $x_1,...,x_n$satisfying:$\sum\limits_{i=1}^n \sqrt{x_i ^2+cx_ix_{i+1}+x_{i+1}^2}=\sqrt{c+2}\left( \sum\limits_{i=1}^n x_i \right)$
holds $c=2$ or $x_1=...=x_n$
When $10^{93}-93$ is expressed as a single whole number, the sum of the digits is
$\text{(A)}\ 10 \qquad \text{(B)}\ 93 \qquad \text{(C)}\ 819 \qquad \text{(D)}\ 826 \qquad \text{(E)}\ 833$
Can two perfect cubes fit between two consecutive perfect squares? In other words, do there exist positive integers $a$, $b$, $n$ such that $n^2 < a^3 < b^3 < (n + 1)^2$?
[i](3 points)[/i]
The equation
$$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.
In the plane, the point $M$ is the midpoint of a line segment $[AB]$ and $\ell$ is an arbitrary line that has no has a common point with the line segment $[AB]$ (and is also not perpendicular to $[AB]$). The points $X$ and $Y$ are the perpendicular projections of $A$ and $B$ onto $\ell$, respectively. Show that the circumscribed circles of triangle $\vartriangle AMX$ and triangle $\vartriangle BMY$ have the same radius.