Found problems: 85335
Solve the equation in natural numbers $$ x+y+z+t=xyzt.
$$
If $f(x)=\frac{1}{x}-\frac{4}{\sqrt{x}}+3$ where $\frac{1}{16}\le x\le 1$, find the range of $f(x)$.
$ \textbf{(A) }-2\le f(x)\le 4 \qquad\textbf{(B) }-1\le f(x)\le 3\qquad\textbf{(C) }0\le f(x)\le 3\qquad\textbf{(D) }-1\le f(x)\le 4\qquad\textbf{(E) }\text{None of the above} $
Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.
[i]Author: Waldemar Pompe, Poland[/i]
A fixed number of people are dividing an inheritance among themselves. An heir will be called poor if he gets less than $\$99$ and rich if he gets more than $\$10 000$ (some heirs may be neither rich nor poor). The total inheritance and the number of heirs are such that the total income of the rich heirs will be no less than that of the poor ones no matter how the inheritance is divided. Prove that the total income of the rich heirs is no less than $100$ times that of the poor ones.
(F Nazarov)
Amelia needs to estimate the quantity $\tfrac ab-c$, where $a$, $b$, and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\tfrac ab-c$?
$\textbf{(A) }\text{She rounds all three numbers up.}$
$\textbf{(B) }\text{She rounds }a\text{ and }b\text{ up, and she rounds }c\text{ down.}$
$\textbf{(C) }\text{She rounds }a\text{ and }c\text{ up, and she rounds }b\text{ down.}$
$\textbf{(D) }\text{She rounds }a\text{ up, and she rounds }b\text{ and }c\text{ down.}$
$\textbf{(E) }\text{She rounds }c\text{ up, and she rounds }a\text{ and }b\text{ down.}$
$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively .
Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$.
Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$.
Prove that the circumcenter of $AIT$ is on the $BC$.
Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation:
$$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$
then $ z_1^6=z_2^6 $
See all the problems from 5-th Kyiv math festival
[url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$
Prove that at least $n - 1$ of these numbers are perfect squares.
Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.
For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.
Two nonnegative integers $a$ and $b$ are [i]tuanis[/i] if the decimal expression of $a+b$ contains only $0$ and $1$ as digits. Let $A$ and $B$ be two infinite sets of non negative integers such that $B$ is the set of all the [i]tuanis[/i] numbers to elements of the set $A$ and $A$ the set of all the [i]tuanis[/i] numbers to elements of the set $B$. Show that in at least one of the sets $A$ and $B$ there is an infinite number of pairs $(x,y)$ such that $x-y=1$.
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?
In cyclic quadrilateral $ABCD$, $\angle DBC = 90^\circ$ and $\angle CAB = 30^\circ$. The diagonals of $ABCD$ meet at $E$. If $\frac{BE}{ED} = 2$ and $CD = 60$, compute $AD$. (Note: a cyclic quadrilateral is a quadrilateral that can be inscribed in a circle.)
Given three planes $\alpha,\beta,\gamma$. Intersection angle between any two planes are all $\theta$.$\alpha\cap\beta=a,\beta\cap\gamma=b,\gamma\cap\alpha=c$.
Given two conditions:
A: $\theta>\frac{\pi}{3}$
B: $a,b,c$ share one point.
$(\text{A})$A is sufficient but unnecessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient and necessary condition of B.
$(\text{D})$None above
Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V.
$ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k.
Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.
Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$?
$
\textbf{(A)}\ 56\sqrt 3
\qquad\textbf{(B)}\ 56 \sqrt 2
\qquad\textbf{(C)}\ 50 \sqrt 2
\qquad\textbf{(D)}\ 84
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Cheryl rolls a fair dice twice. If the dice's six faces are numbered by $1$, $2$, $3$, $4$, $5$, $6$, what is the probability that the number on one of her rolls is a divisor of the number on the other roll?
$\textbf{(A) }\dfrac29\qquad\textbf{(B) }\dfrac5{18}\qquad\textbf{(C) }\dfrac49\qquad\textbf{(D) }\dfrac12\qquad\textbf{(E) }\dfrac{11}{18}$
A sequence $a_1, a_2,\dots, a_n$ of positive integers is [i]alagoana[/i], if for every $n$ positive integer, one have these two conditions
I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$
II- The number $a_n$ is the $n$-power of a positive integer.
Find all the sequence(s) [i]alagoana[/i].
Let $n$ be a positive integer and let $t$ be an integer. $n$ distinct integers are written on a table. Bob, sitting in a room nearby, wants to know whether there exist some of these numbers such that their sum is equal to $t$. Alice is standing in front of the table and she wants to help him. At the beginning, she tells him only the initial sum of all numbers on the table. After that, in every move he says one of the $4$ sentences:
$i.$ Is there a number on the table equal to $k$?
$ii.$ If a number $k$ exists on the table, erase him.
$iii.$ If a number $k$ does not exist on the table, add him.
$iv.$ Do the numbers written on the table can be arranged in two sets with equal sum of elements?
On these questions Alice answers yes or no, and the operations he says to her she does (if it is possible) and does not tell him did she do it. Prove that in less than $3n$ moves, Bob can find out whether there exist numbers initially written on the board such that their sum is equal to $t$
The eight pointed star is a popular quilting pattern. What percent of the entire 4-by-4 grid is covered by the star?
$(A)40$ $~~~$ $(B)50$ $~~~$ $(C)60$ $~~~$ $(D)75$ $~~~$ $(E)80$
1) Is it possible to place five circles on the plane in such way that each circle has exactly 5 common points with other circles?
2) Is it possible to place five circles on the plane in such way that each circle has exactly 6 common points with other circles?
3) Is it possible to place five circles on the plane in such way that each circle has exactly 7 common points with other circles?
Find all functions $ f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $ x$, $ y$ $ \in\mathbb{R}$,
\[ f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y)\]
holds.
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$.
Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.
[i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$.
Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.
There are nine boxes. In the first there is $1$ stone, in the second there are $2$ stones, in the third there are $3$ stones, and thus continuing, in the eighth there are $8$ stones and in the ninth there are $9$ stones. The allowed operation is to remove the same number of stones from two different boxes and place them in a third box. The goal is that all stones are in a single box. Describe how to do it with the minimum number of operations allowed.
Explain why it is impossible to achieve it with fewer operations.