Found problems: 85335
2002 Austrian-Polish Competition, 4
For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?
2018 Austria Beginners' Competition, 4
For a positive integer $n$ we denote by $d(n)$ the number of positive divisors of $n$ and by $s(n)$ the sum of these divisors. For example, $d(2018)$ is equal to $4$ since $2018$ has four divisors $(1, 2, 1009, 2018)$ and $s(2018) = 1 + 2 + 1009 + 2018 = 3030$.
Determine all positive integers $x$ such that $s(x) \cdot d(x) = 96$.
(Richard Henner)
2005 China Team Selection Test, 2
Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.
1971 AMC 12/AHSME, 2
If $b$ men take $c$ days to lay $f$ bricks, then the number of days it will take $c$ men working at the same rate to lay $b$ bricks, is
$\textbf{(A) }fb^2\qquad\textbf{(B) }b/f^2\qquad\textbf{(C) }f^2/b\qquad\textbf{(D) }b^2/f\qquad \textbf{(E) }f/b^2$
2010 IFYM, Sozopol, 6
In $\Delta ABC$ $(AB>BC)$ $BM$ and $BL$ $(M,L\in AC)$ are a median and an angle bisector respectively. Let the line through $M$, parallel to $AB$, intersect $BL$ in point $D$ and the line through $L$, parallel to $BC$, intersect $BM$ in point $E$. Prove that $DE\perp BL$.
2015 BMT Spring, Tie 2
The unit square $ABCD$ has $E$ as midpoint of $AD$ and a circle of radius $r$ tangent to $AB$, $BC$, and $CE$. Determine $r$.
2015 AMC 10, 14
Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$?
$\textbf{(A) } 15
\qquad\textbf{(B) } 15.5
\qquad\textbf{(C) } 16
\qquad\textbf{(D) } 16.5
\qquad\textbf{(E) } 17
$
2010 ISI B.Stat Entrance Exam, 8
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]
1968 Yugoslav Team Selection Test, Problem 1
Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.
2014 BMT Spring, 5
Determine
$$\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}$$
2022 Azerbaijan IMO TST, 1
Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.
2019 Saudi Arabia Pre-TST + Training Tests, 3.3
Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.
2004 Purple Comet Problems, 4
If the numbers $2a + 2$ and $2b + 2$ add up to $2004$, find the sum of the numbers $\frac{a}{2} - 2$ and $\frac{b}{2} - 2$
2017 Istmo Centroamericano MO, 4
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.
2018 Slovenia Team Selection Test, 2
Ana and Bojan are playing a game: Ana chooses positive integers $a$ and $b$ and each one gets $2016$ pieces of paper, visible to both - Ana gets the pieces with the numbers $a+1$, $a+2$, $\ldots$, $a+2016$ and Bojan gets the pieces with the numbers $b+1$, $b+2$, $\ldots$, $b+2016$ on them. Afterwards, one of them writes the number $a+b$ on the board. In every move, Ana chooses one of her pieces of paper and hands it to Bojan who chooses one of his own, writes their sum on the board and removes them both from the game. When they run out of pieces, they multiply the numbers on the board together. If the result has the same remainder than $a+b$ when divided by $2017$, Bojan wins, otherwise, Ana wins. Who has the winning strategy?
1998 Canada National Olympiad, 4
Let $ABC$ be a triangle with $\angle{BAC} = 40^{\circ}$ and $\angle{ABC}=60^{\circ}$. Let $D$ and $E$ be the points lying on the sides $AC$ and $AB$, respectively, such that $\angle{CBD} = 40^{\circ}$ and $\angle{BCE} = 70^{\circ}$. Let $F$ be the point of intersection of the lines $BD$ and $CE$. Show that the line $AF$ is perpendicular to the line $BC$.
1999 National Olympiad First Round, 35
Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k \minus{} 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle?
$\textbf{(A)}\ 14 \qquad\textbf{(B)}\ 53 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 79 \qquad\textbf{(E)}\ 156$
2018 Purple Comet Problems, 14
A complex number $z$ whose real and imaginary parts are integers satis fies $\left(Re(z) \right)^4 +\left(Re(z^2)\right)^2 + |z|^4 =(2018)(81)$, where $Re(w)$ and $Im(w)$ are the real and imaginary parts of $w$, respectively. Find $\left(Im(z) \right)^2$
.
2018 Dutch BxMO TST, 4
In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.
2000 Greece Junior Math Olympiad, 3
On a past Mathematical Olympiad the maximum possible score on a problem was 5. The average score of boys was 4, the average score of girls was 3.25, and the overall average score was 3.60. Find the total number of participants, knowing that it was in the range from 31 to 50.
1996 Chile National Olympiad, 6
Two circles, $C$ and $K$, are secant at $A$ and $B$. Let $P$ be a point on the arc $AB$ of $C$. Lines $PA$ and $PB$ intersect $K$ again at $R$ and $S$ respectively. Let $P'$ be another point at same arc as $P$, so that lines $P'A$ and $P'B$ again intersect $K$ at $R'$ and $S'$, respectively. Prove that the arcs $RS$ and $R'S'$ have equal measures.
[img]https://cdn.artofproblemsolving.com/attachments/2/4/88693c36159179fb2b098b671a2f8281b37aae.png[/img]
2023 Math Prize for Girls Olympiad, 2
The two cats Fitz and Will play the following game. On a blackboard is written the expression
\[
x^{100} + {\square} x^{99} + {\square} x^{98} + {\square} x^{97} + \dots + {\square } x^2 + {\square} x +1.
\]
Both cats take alternate turns replacing one $\square$ with a $0$ or $1$, with Fitz going first, until (after 99 turns) all the blanks have been filled. If the resulting polynomial obtained has a real root, then Will wins, otherwise Fitz wins. Determine, with proof, which player has a winning strategy.
2017 Vietnamese Southern Summer School contest, Problem 4
Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$.
1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$.
2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.
2003 Purple Comet Problems, 20
In how many ways can we form three teams of four players each from a group of $12$ participants?
PEN O Problems, 39
Find the smallest positive integer $n$ for which there exist $n$ different positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying [list] [*] $\text{lcm}(a_1,a_2,\cdots,a_n)=1985$,[*] for each $i, j \in \{1, 2, \cdots, n \}$, $gcd(a_i,a_j)\not=1$, [*] the product $a_{1}a_{2} \cdots a_{n}$ is a perfect square and is divisible by $243$, [/list] and find all such $n$-tuples $(a_{1}, \cdots, a_{n})$.