Found problems: 85335
Let $ABCD$ be a square of the plane $P$. Define the minimum and the maximum the value of the function $f: P \to R$ is given by $f (P) =\frac{PA + PB}{PC + PD}$
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
$ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ have radii of $20$ and $16$, respectively, find $\sqrt{1+BC/BD}$.
Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$
Two isosceles triangles of the same area are located as shown in the figure. Find the angle $x$.
[img]https://cdn.artofproblemsolving.com/attachments/a/6/f7dbfd267274781b67a5f3d5a9036fb2905156.png[/img]
How many subsets of $\{1, 2, \cdots, 10\}$ are there that don't contain $2$ consecutive integers?
Find all positive integers $m$ with the next property:
If $d$ is a positive integer less or equal to $m$ and it isn't coprime to $m$ , then there exist positive integers $a_{1}, a_{2}$,. . ., $a_{2019}$ (where all of them are coprimes to $m$) such that
$m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}$
is a perfect power.
Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $$\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$$
Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.
A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle.
[img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]
The sum of the squares of the roots of the equation $x^2+2hx=3$ is $10$. The absolute value of $h$ is equal to
$\textbf{(A) }-1\qquad\textbf{(B) }\textstyle\frac{1}{2}\qquad\textbf{(C) }\textstyle\frac{3}{2}\qquad\textbf{(D) }2\qquad \textbf{(E) }\text{None of these}$
How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$
Suppose two positive real numbers are given. Prove that if their sum is less than their product then their sum is greater than four.
(N Vasiliev, Moscow)
Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular drawn from point $F$ on $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$.
KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse
Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.
Nine numbers are arranged in a square table:
$a_1 \,\,\, a_2 \,\,\,a_3$
$b_1 \,\,\,b_2 \,\,\,b_3$
$c_1\,\,\, c_2 \,\,\,c_3$ .
It is known that the six numbers obtained by summing the rows and columns of the table are equal:
$a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ .
Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns:
$a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ .
(V Proizvolov)
We have a $2007 \times 2007$ square table fi lled with nonnegative integers. For each entry of $0$ in the table, the sum of the elements that are in the same row or column as that entry is at least $2007$. Find the minimum sum of all the elements of such a table.
Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.
Find all real numbers with the following property: the difference of its cube and
its square is equal to the square of the difference of its square and the number itself.
Let $\omega_1$ and $\omega_2$ be two circles such that the center $O$ of $\omega_2$ lies in $\omega_1$. Let $C$ and $D$ be the two intersection points of the circles. Let $A$ be a point on $\omega_1$ and let $B$ be a point on $\omega_2$ such that $AC$ is tangent to $\omega_2$ in C and BC is tangent to $\omega_1$ in $C$. The line segment $AB$ meets $\omega_2$ again in $E$ and also meets $\omega_1$ again in F. The line $CE$ meets $\omega_1$ again in $G$ and the line $CF$ meets the line $GD$ in $H$. Prove that the intersection point of $GO$ and $EH$ is the center of the circumcircle of the triangle $DEF$.
Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.
In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)
In four functions $y=\sin|x|,y=\cos|x|,y=|\cot x|,y=\lg|\sin x|$, which one is even function, and increases on $\left(0,\frac{\pi}{2}\right)$, with period of $\pi$?
$\text{(A)}y=\sin|x|\qquad\text{(B)}y=\cos|x|\qquad\text{(C)}y=|\cot x|\qquad\text{(D)}y=\lg|\sin x|$