Found problems: 85335
For each positive integer $n$, define $A(n)$ to be the sum of its divisors, and $B(n)$ to be the sum of products of pairs of its divisors. For example,
\[A(10)=1+2+5+10=18\]
\[B(10)=1\cdot 2+1\cdot 5+1\cdot 10+2\cdot 5+2\cdot 10+5\cdot 10=97\]
Find all positive integers $n$ for which $A(n)$ divides $B(n)$.
A $3 \times 3$ square is filled with numbers: $a, b, c, d, e, f, g, h, i$ in the following way: [img]https://cdn.artofproblemsolving.com/attachments/8/9/737c41e9d0dbfdc81be1b986b8e680290db55e.png[/img]
Given that the square is magic (sums of the numbers in each row, column and each of two diagonals are the same), show that
a) $2(a + c + g + i) = b + d + f + h + 4e$. (3)
b) $2(a^3 + c^3 + g^3 + i^3) = b^3 + d^3 + f^3 + h^3 + 4e^3$. (3)
Prove that for all positive real numbers $x, y$ and $z$, the double inequality $$0 < \frac{1}{x + y + z + 1} -\frac{1}{(x + 1)(y + 1)(z + 1)} \le \frac18$$ holds. When does equality hold in the right inequality?
[i](Walther Janous)[/i]
Let $\Gamma_1$ and $\Gamma_2$ be two circles with radii $r_1$ and $r_2,$ respectively, where $r_1>r_2.$ Suppose $\Gamma_1$ and $\Gamma_2$ intersect at two distinct points $A$ and $B.$ A point $C$ is selected on ray $\overrightarrow{AB},$ past $B,$ and the tangents to $\Gamma_1$ and $\Gamma_2$ from $C$ are marked as points $P$ and $Q,$ respectively. Suppose that $\Gamma_2$ passes through the center of $\Gamma_1$ and that points $P, B, Q$ are collinear in that order, with $PB=3$ and $QB=2.$ What is the length of $AB?$
[i]Proposed by Kyle Lee[/i]
Let $ \vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $ [0,1]$. Define \[ h(x)\equal{} \frac1n \# \{k: \; \vartheta_k<x\ \}.\] Prove that the probability that there is an $ x_0 \in (0,1)$ such that $ h(x_0)\equal{}x_0$, is equal to $ 1\minus{} \frac1n.$
[i]G. Tusnady[/i]
The integers $1, 2,. . . , n$ are arranged in order so that each value is strictly larger than all values above or is strictly less than all values previous. In how many ways can this be done?
Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.
Let $\triangle ABC$ be such that the midpoint of $BC$ is $D$. Let $E$ be the point on the opposite side of $AC$ as $B$ on the circumcircle of $\triangle ABC$ such that $\angle DEA = \angle DEC$ and let $\omega$ be the circumcircle of $\triangle CED$. If $\omega$ intersects $AE$ at $X$ and the tangent to $\omega$ at $D$ intersects $AB$ at $Y$, show that $XY$ is parallel to $BC$.
[i]Proposed by Taco12[/i]
Triangle $ABC$ is given for which $BC = AC + \frac{1}{2}AB$. The point $P$ divides $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2\angle CPA$.
A bottle contains $5$ gallons of a $10\%$ solution of oil. How many gallons of pure oil must be added to make a $30\%$ oil solution? (round to the nearest hundredth)
Let $ABC$ be an acute, non-isosceles triangle with altitude $AD$ ($D \in BC$), $M$ is the midpoint of $AD$ and $O$ is the circumcenter. Line $AO$ meets $BC$ at $K$ and circle of center $K$, radius $KA$ cuts $AB,AC$ at $E, F$ respectively. Prove that $AO$ bisects $EF$.
Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$.
For which values of n does there exist a polyhedron having $n$ edges?
Show that [0,1] cannot be partitioned into two disjoints sets A and B such that B=A+a for some real a.
On the sides $AB$ and $AC$ of the right $\triangle ABC$ ($\angle A=90^{\circ}$) are chosen points $C_{1}$ and $B_{1}$ respectively. Prove that if $M=CC_{1}\cap BB_{1}$ and $AC_{1}=AB_{1}=AM$, then $[AB_{1}MC_{1}]+[AB_{1}C_{1}]=[BMC]$.
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \ge 2017$, the integer $P(n)$ is positive and $S(P(n)) = P(S(n))$.
Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?
$\textbf{(A)}\ 15 \qquad
\textbf{(B)}\ 16\qquad
\textbf{(C)}\ 17\qquad
\textbf{(D)}\ 21\qquad
\textbf{(E)}\ 37$
A $ 3\times3\times3$ cube composed of $ 27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $ 3\times3\times1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $ 45^\circ$ angle with the horizontal plane.
Let $ m$ and $ n$ be positive integers with $ m > n \geq 2.$ Set $ S \equal{} \{1, 2, \ldots, m\},$ and $ T \equal{} \{a_l, a_2, \ldots, a_n\}$ is a subset of S such that every number in $ S$ is not divisible by any two distinct numbers in $ T.$ Prove that
\[ \sum^n_{i \equal{} 1} \frac {1}{a_i} < \frac {m \plus{} n}{m}.
\]
Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and
\[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]
The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon.
(a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.)
(b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter.
(c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths.
It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle.
(d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon
can be increased without changing the perimeter.
(e) Prove that the polygon of maximum area must be equiangular, or have all angles equal.
(f ) Prove that the polygon of maximum area is a regular polygon.
PS. You had better use hide for answers.
The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played?
A circle passing through $B$ and $C$ meets the side $[AB]$ of $\triangle ABC$ at $D$, and $[AC]$ at $E$. The circumcircle of $\triangle ACD$ intersects with $BE$ at a point $F$ outside $[BE]$. If $|AD| = 4, |BD|= 8$, then what is $|AF|$?
$\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2\sqrt6 \qquad\textbf{(C)}\ 4\sqrt6 \qquad\textbf{(D)}\ \sqrt6 \qquad\textbf{(E)}\ \text{None}$
Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define $$f(A)=\sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.$$If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the remainder when the sum of the distinct prime factors of $\mathbb{E}[f(A)]$ is divided by $1000$.
[i]Proposed by [b]FedeX333X[/b][/i]
Determine all tuples of integers $(a,b,c)$ such that:
$$(a-b)^3(a+b)^2 = c^2 + 2(a-b) + 1$$