Found problems: 85335
Let $ \{ f_n \}$ be a sequence of Lebesgue-integrable functions on $ [0,1]$ such that for any Lebesgue-measurable subset $ E$ of $ [0,1]$ the sequence $ \int_E f_n$ is convergent. Assume also that $ \lim_n f_n\equal{}f$ exists almost everywhere. Prove that $ f$ is integrable and $ \int_E f\equal{}\lim_n \int_E f_n$. Is the assertion also true if $ E$ runs only over intervals but we also assume $ f_n \geq 0 ?$ What happens if $ [0,1]$ is replaced by $ [0,\plus{}\infty) ?$
[i]J. Szucs[/i]
At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
$\textbf{(A) }23 \qquad \textbf{(B) }32 \qquad \textbf{(C) }37 \qquad \textbf{(D) }41 \qquad \textbf{(E) }64$
For a set \(S\) of positive integers and a positive integer \(n\), consider the game of [i]\((n,S)\)-nim[/i], which is as follows. A pile starts with \(n\) watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of \(S\). The last player to move wins. Let \(f(S)\) denote the set of positive integers \(n\) for which Deric has a winning strategy in \((n,S)\)-nim.
Let \(T\) be a set of positive integers. Must the sequence \[T, \; f(T), \; f(f(T)), \;\ldots\] be eventually constant?
[i]Proposed by Brandon Wang and Edward Wan[/i]
Let $(X,d)$ be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that $X$ has exactly one element.
If $p$, $q$ and $M$ are positive numbers and $q<100$, then the number obtained by increasing $M$ by $p\%$ and decreasing the result by $q\%$ exceeds $M$ if and only if
$\text{(A)}\ p>q ~~ \text{(B)}\ p>\frac{q}{100-q} ~~ \text{(C)}\ p>\frac{q}{1-q} ~~ \text{(D)}\ p>\frac{100q}{100+q} ~~ \text{(E)}\ p>\frac{100q}{100-q}$
Prove that if every subspace of a Hausdorff space $X$ is $\sigma$-compact, then $X$ is countable.
Given are arbitrary integers $a,b,p$. Prove that there always exist relatively prime integers $k$ and $\ell$ such that $ak+b\ell$ is divisible by $p$.
There are $n > 1$ points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?
Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$.
$ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$
Evaluate $$\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \min (n, k) \left( \frac12 \right)^n\left( \frac13 \right)^k$$
Ten students are standing in a line. A teacher wants to place a hat on each student. He has two colors of hats, red and white, and he has $10$ hats of each color. Determine the number of ways in which the teacher can place hats such that among any set of consecutive students, the number of students with red hats and the number of students with blue hats differ by at most $2$
Let the interior point $P$ of the convex quadrilateral $ABCD$ be such that $$|\angle PAD| = |\angle ADP| = |\angle CBP| = |\angle PCB| = |\angle CPD|.$$ Let $O$ be the center of the circumcircle of the triangle $CPD$. Prove that $|OA| = |OB|$.
A $ 3 \times 3 \times 3$ cube is formed by gluing together 27 standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the $ 3 \times 3 \times 3$ cube is
$ \text{(A)}\ 60 \qquad
\text{(B)}\ 72 \qquad
\text{(C)}\ 84 \qquad
\text{(D)}\ 90 \qquad
\text{(E)}\ 96$
For a pair $ A \equal{} (x_1, y_1)$ and $ B \equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \equal{} |x_1 \minus{} x_2| \plus{} |y_1 \minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \leq 2$. Determine the maximum number of harmonic pairs among 100 points in the plane.
In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.
[u]Set 7[/u]
[b]G19.[/b] How many ordered triples $(x, y, z)$ with $1 \le x, y, z \le 50$ are there such that both $x + y + z$ and $xy + yz + zx$ are divisible by$ 6$?
[b]G20.[/b] Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. If $D$ is the foot of the perpendicular from $A$ to $BC$, then $AH = 8$ and $HD = 3$. If $\angle AOH = 90^o$, find $BC^2$.
[b]G21.[/b] Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly $12$ times is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[u]Set 8[/u]
[b]G22.[/b] Let $f$ be a function defined by $f(1) = 1$ and $$f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2,$$ where $p$ is the least prime dividing $n$, for all integers $n \ge 2$. Find $f(2022)$.
[b]G23.[/b] Jessica has $15$ balls numbered $1$ through $15$. With her left hand, she scoops up $2$ of the balls. With her right hand, she scoops up $2$ of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[b]G24.[/b] Let $ABCD$ be a cyclic quadrilateral such that its diagonal $BD = 17$ is the diameter of its circumcircle. Given $AB = 8$, $BC = CD$, and that a line $\ell$ through A intersects the incircle of $ABD$ at two points $P$ and $Q$, the maximum area of $CP Q$ can be expressed as a fraction $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
[u]Set 9[/u]
[i]This set consists of three estimation problems, with scoring schemes described.[/i]
[b]G25.[/b] Estimate $N$, the total number of participants (in person and online) at MOAA this year. An estimate of $e$ gets a total of max $ \left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)$ points.
[b]G26.[/b] If $A$ is the the total number of in person participants at MOAA this year, and $B$ is the total number of online participants at MOAA this year, estimate $N$, the product $AB$. An estimate of $e$ gets a total of max $(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )$ points.
[b]G27.[/b] Estimate $N$, the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max $(0, 30 - \lceil 7 log5(|N - E|)\rceil )$ points.
PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 4-6 [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Consider skew lines $PP'$, $QQ'$ and points $X$, $Y$ lying on them, respectively. Initially, we have $X=P$, $Y=Q$. Both points $X$, $Y$ start moving simultaneously along the rays $PP'$, $QQ'$ with the speeds $c_1$, $c_2$, respectively. Show that midpoint $Z$ of segment $XY$ always lies on a fixed ray $RR'$, where $R$ is midpoint of $PQ$.
Quadratic polynomials $P(x)$ and $Q(x)$ have leading coefficients of $2$ and $-2$, respectively. The graphs of both polynomials pass through the two points $(16,54)$ and $(20,53)$. Find ${P(0) + Q(0)}$.
Let $ABC$ be an acute-angled triangle, with $\angle B \neq \angle C$ . Let $M$ be the midpoint of side $BC$, and $E,F$ be the feet of the altitude from $B,C$ respectively. Denote by $K,L$ the midpoints of segments $ME,MF$, respectively. Suppose $T$ is a point on the line $KL$ such that $AT//BC$.
Prove that $TA=TM$ .
Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?
$ \textbf{(A)}\ 1/4 \qquad \textbf{(B)}\ 1/3 \qquad \textbf{(C)}\ 3/8 \qquad \textbf{(D)}\ 2/5 \qquad \textbf{(E)}\ 1/2$
A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.
Prove that the equation $4^x +6^x =9^x$ has no rational solutions.
Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers
\[
a_n = 4^n c + \frac{4^n - (-1)^n}{5}
\]
contains at least one perfect square.
Let $n$ be an integer with $n \ge 2$. Show that $\phi(2^{n}-1)$ is divisible by $n$.
Polynomial $P(x)$ with degree $n \geq 3$ has $n$ real roots $x_1 < x_2 < x_3 <...< x_n$, such that $x_2-x_1<x_3-x_2<....<x_n-x_{n-1}$. Prove that the maximum of the function $y=|P(x)|$ where $x$ is on the interval $[ x_1, x_n ]$, is in the interval $[x_n-1, x_n]$.