Found problems: 85335
DMM Team Rounds, 2011
[b]p1.[/b] How many primes $p < 100$ satisfy $p = a^2 + b^2$ for some positive integers $a$ and $b$?
[b]p2. [/b] For $a < b < c$, there exists exactly one Pythagorean triple such that $a + b + c = 2000$. Find $a + c - b$.
[b]p3.[/b] Five points lie on the surface of a sphere of radius $ 1$ such that the distance between any two points is at least $\sqrt2$. Find the maximum volume enclosed by these five points.
[b]p4.[/b] $ABCDEF$ is a convex hexagon with $AB = BC = CD = DE = EF = FA = 5$ and $AC = CE = EA = 6$. Find the area of $ABCDEF$.
[b]p5.[/b] Joe and Wanda are playing a game of chance. Each player rolls a fair $11$-sided die, whose sides are labeled with numbers $1, 2, ... , 11$. Let the result of the Joe’s roll be $X$, and the result of Wanda’s roll be $Y$ . Joe wins if $XY$ has remainder $ 1$ when divided by $11$, and Wanda wins otherwise. What is the probability that Joe wins?
[b]p6.[/b] Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number $n$ is divisible by $3$, he replaces $n$ with $\frac{n}{3} + 2$, and otherwise he replaces $n$ with $\lfloor 3 \log_3 n \rfloor$. If he starts with the number $3^{2011}$, what number will he have after $2011$ steps?
Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
[b]p7.[/b] Define a sequence an of positive real numbers with a$_1 = 1$, and $$a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}.$$
What is $a_{2011}$?
[b]p8.[/b] A set $S$ of positive integers is called good if for any $x, y \in S$ either $x = y$ or $|x - y| \ge 3$. How many subsets of $\{1, 2, 3, ..., 13\}$ are good? Include the empty set in your count.
[b]p9.[/b] Find all pairs of positive integers $(a, b)$ with $a \le b$ such that $10 \cdot lcm \, (a, b) = a^2 + b^2$. Note that $lcm \,(m, n)$ denotes the least common multiple of $m$ and $n$.
[b]p10.[/b] For a natural number $n$, $g(n)$ denotes the largest odd divisor of $n$. Find $$g(1) + g(2) + g(3) + ... + g(2^{2011})$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 Iranian Geometry Olympiad, 5
Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$. Points $E$, $F$ lie on its side $AC$, $AB$, respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$, $S$ be the intersections of $EF$ with the shorter arcs $AB$, $AC$ of $\omega$, respectively. Suppose $K$, $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$, respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$, respectively, such that $PK$ and $QL$ are perpendicular to $BC$. Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$.
[i]Proposed by Mehran Talaei[/i]
2005 China Team Selection Test, 3
Let $n$ be a positive integer, set $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$. For any two elements $a=(a_1,a_2,\cdots,a_{2^n})$ and $b=(b_1,b_2,\cdots,b_{2^n})$ of $S_n$, define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call $A \subseteq S_n$ a $\textsl{Good Subset}$ if $d(a,b) \geq 2^{n-1}$ holds for any two distinct elements $a$ and $b$ of $A$. How many elements can the $\textsl{Good Subset}$ of $S_n$ at most have?
2022 CCA Math Bonanza, L3.1
Kongol rolls two fair 6-sided die. The probability that one roll is a divisor of the other can be expressed as $\frac{p}{q}$. Determine $p+q$.
[i]2022 CCA Math Bonanza Lightning Round 3.1[/i]
Mid-Michigan MO, Grades 10-12, 2014
[b]p1.[/b] The length of the side $AB$ of the trapezoid with bases $AD$ and $BC$ is equal to the sum of lengths $|AD|+|BC|$. Prove that bisectors of angles $A$ and $B$ do intersect at a point of the side $CD$.
[b]p2.[/b] Polynomials $P(x) = x^4 + ax^3 + bx^2 + cx + 1$ and $Q(x) = x^4 + cx^3 + bx^2 + ax + 1$ have two common roots. Find these common roots of both polynomials.
[b]p3.[/b] A girl has a box with $1000$ candies. Outside the box there is an infinite number of chocolates and muffins. A girl may replace:
$\bullet$ two candies in the box with one chocolate bar,
$\bullet$ two muffins in the box with one chocolate bar,
$\bullet$ two chocolate bars in the box with one candy and one muffin,
$\bullet$ one candy and one chocolate bar in the box with one muffin,
$\bullet$ one muffin and one chocolate bar in the box with one candy.
Is it possible that after some time it remains only one object in the box?
[b]p4.[/b] There are $9$ straight lines drawn in the plane. Some of them are parallel some of them intersect each other. No three lines do intersect at one point. Is it possible to have exactly $17$ intersection points?
[b]p5.[/b] It is known that $x$ is a real number such that $x+\frac{1}{x}$ is an integer. Prove that $x^n+\frac{1}{x^n}$ is an integer for any positive integer $n$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2019 Saudi Arabia JBMO TST, 2
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$.
Prove that $\angle A = 2\angle B$ if and only if $AC = 2MD$
1999 Hong kong National Olympiad, 1
Find all positive rational numbers $r\not=1$ such that $r^{\frac{1}{r-1}}$ is rational.
2015 Dutch IMO TST, 1
In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.
2016 Korea USCM, 7
$M$ is a postive real and $f:[0,\infty)\to[0,M]$ is a continuous function such that
$$\int_0^\infty (1+x)f(x) dx<\infty$$
Then, prove the following inequality.
$$\left(\int_0^\infty f(x) dx \right)^2 \leq 4M \int_0^\infty x f(x) dx$$
(@below, Thank you. I fixed.)
1985 Polish MO Finals, 4
$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$
1997 Estonia Team Selection Test, 2
A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.
1993 IMO Shortlist, 5
$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. [hide="For example"] P could be $3*5$, but not $3^2*5$.[/hide] Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$.
[b]Original Statement:[/b]
Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which \[ ax^2 + 2bxy + cy^2 = n. \] Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.
2018 Harvard-MIT Mathematics Tournament, 10
One million [i]bucks [/i] (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are [i]honest bucks[/i], who always tell the truth, and the remaining bucks are [i]dishonest bucks[/i], who never tell the truth.
Each of the million [i]bucks [/i] claims that “at most one of my neighboring bucks is an [i]honest buck[/i].” A pair of [i]neighboring bucks[/i] is said to be [i]buckaroo[/i] if exactly one of them is an [i]honest buck[/i] . What is the minimum possible number of [i]buckaroo [/i] pairs in the grid?
Note: Two [i]bucks [/i] are considered to be [i]neighboring [/i] if their cells $(x_1, y_1)$ and $(x_2, y_2)$ satisfy either:
$x_1 = x_2$ and $y_1 - y_2 \equiv \pm1$ (mod $1000$), or $x_1 - x_2 \equiv \pm 1$ (mod $1000$) and $y_1 = y_2$.
2013 Brazil Team Selection Test, 4
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.
2001 Federal Math Competition of S&M, Problem 4
There are $n$ coins in the pile. Two players play a game by alternately performing a move. A move consists of taking $5,7$ or $11$ coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if:
(a) $n=2001$;
(b) $n=5000$?
2005 Pan African, 3
Let $ABC$ be a triangle and let $P$ be a point on one of the sides of $ABC$. Construct a line passing through $P$ that divides triangle $ABC$ into two parts of equal area.
PEN S Problems, 16
Show that if $a$ and $b$ are positive integers, then \[\left( a+\frac{1}{2}\right)^{n}+\left( b+\frac{1}{2}\right)^{n}\] is an integer for only finitely many positive integer $n$.
2002 Mediterranean Mathematics Olympiad, 1
Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.
2021 Iranian Geometry Olympiad, 1
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point
$E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that
$BE \perp HD$.
[i]Proposed by Tran Quang Hung - Vietnam[/i]
2005 Purple Comet Problems, 22
Let $d_k$ be the greatest odd divisor of $k$ for $k = 1, 2, 3, \ldots$. Find $d_1 + d_2 + d_3 + \ldots + d_{1024}$.
VMEO III 2006 Shortlist, N11
Prove that the composition of the sets of one of the following two forms is finite:
(a) $2^{2^n}+1$
(b) $6^{2^n}+1$
2016 India PRMO, 6
Suppose a circle $C$ of radius $\sqrt2$ touches the $Y$ -axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects on a point $P$ on the circumference of $C$, and after reflection, the reflected ray $L'$ becomes parallel to the $Y$ -axis. Find the distance between the ray $L$ and the $X$-axis.
2007 All-Russian Olympiad, 3
Two players by turns draw diagonals in a regular $(2n+1)$-gon ($n>1$). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy?
[i]K. Sukhov[/i]
2012 AMC 8, 16
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
$\textbf{(A)}\hspace{.05in}76531 \qquad \textbf{(B)}\hspace{.05in}86724 \qquad \textbf{(C)}\hspace{.05in}87431 \qquad \textbf{(D)}\hspace{.05in}96240 \qquad \textbf{(E)}\hspace{.05in}97403 $
1951 AMC 12/AHSME, 8
The price of an article is cut $ 10\%$. To restore it to its former value, the new price must be increased by:
$ \textbf{(A)}\ 10\% \qquad\textbf{(B)}\ 9\% \qquad\textbf{(C)}\ 11\frac {1}{9}\% \qquad\textbf{(D)}\ 11\% \qquad\textbf{(E)}\ \text{none of these answers}$