Found problems: 85335
2005 AIME Problems, 11
A semicircle with diameter $d$ is contained in a square whose sides have length $8$. Given the maximum value of $d$ is $m- \sqrt{n}$, find $m+n$.
ABMC Accuracy Rounds, 2022
[b]p1.[/b] Let $X = 2022 + 022 + 22 + 2$. When $X$ is divided by $22$, there is a remainder of $R$. What is the value of $R$?
[b]p2.[/b] When Amy makes paper airplanes, her airplanes fly $75\%$ of the time. If her airplane flies, there is a $\frac56$ chance that it won’t fly straight. Given that she makes $80$ airplanes, what is the expected number airplanes that will fly straight?
[b]p3.[/b] It takes Joshua working alone $24$ minutes to build a birdhouse, and his son working alone takes $16$ minutes to build one. The effective rate at which they work together is the sum of their individual working rates. How long in seconds will it take them to make one birdhouse together?
[b]p4.[/b] If Katherine’s school is located exactly $5$ miles southwest of her house, and her soccer tournament is located exactly $12$ miles northwest of her house, how long, in hours, will it take Katherine to bike to her tournament right after school given she bikes at $0.5$ miles per hour? Assume she takes the shortest path possible.
[b]p5.[/b] What is the largest possible integer value of $n$ such that $\frac{4n+2022}{n+1}$ is an integer?
[b]p6.[/b] A caterpillar wants to go from the park situated at $(8, 5)$ back home, located at $(4, 10)$. He wants to avoid routes through $(6, 7)$ and $(7, 10)$. How many possible routes are there if the caterpillar can move in the north and west directions, one unit at a time?
[b]p7.[/b] Let $\vartriangle ABC$ be a triangle with $AB = 2\sqrt{13}$, $BC = 6\sqrt2$. Construct square $BCDE$ such that $\vartriangle ABC$ is not contained in square $BCDE$. Given that $ACDB$ is a trapezoid with parallel bases $\overline{AC}$, $\overline{BD}$, find $AC$.
[b]p8.[/b] How many integers $a$ with $1 \le a \le 1000$ satisfy $2^a \equiv 1$ (mod $25$) and $3^a \equiv 1$ (mod $29$)?
[b]p9.[/b] Let $\vartriangle ABC$ be a right triangle with right angle at $B$ and $AB < BC$. Construct rectangle $ADEC$ such that $\overline{AC}$,$\overline{DE}$ are opposite sides of the rectangle, and $B$ lies on $\overline{DE}$. Let $\overline{DC}$ intersect $\overline{AB}$ at $M$ and let $\overline{AE}$ intersect $\overline{BC}$ at $N$. Given $CN = 6$, $BN = 4$, find the $m+n$ if $MN^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$.
[b]p10.[/b] An elimination-style rock-paper-scissors tournament occurs with $16$ players. The $16$ players are all ranked from $1$ to $16$ based on their rock-paper-scissor abilities where $1$ is the best and $16$ is the worst. When a higher ranked player and a lower ranked player play a round, the higher ranked player always beats the lower ranked player and moves on to the next round of the tournament. If the initial order of players are arranged randomly, and the expected value of the rank of the $2$nd place player of the tournament can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$ what is the value of $m+n$?
[b]p11.[/b] Estimation (Tiebreaker) Estimate the number of twin primes (pairs of primes that differ by $2$) where both primes in the pair are less than $220022$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
KoMaL A Problems 2019/2020, A. 771
Let $\omega$ denote the incircle of triangle $ABC,$ which is tangent to side $BC$ at point $D.$ Let $G$ denote the second intersection of line $AD$ and circle $\omega.$ The tangent to $\omega$ at point $G$ intersects sides $AB$ and $AC$ at points $E$ and $F$ respectively. The circumscribed circle of $DEF$ intersects $\omega$ at points $D$ and $M.$ The circumscribed circle of $BCG$ intersects $\omega$ at points $G$ and $N.$ Prove that lines $AD$ and $MN$ are parallel.
[i]Proposed by Ágoston Győrffy, Remeteszőlős[/i]
2015 Balkan MO, 4
Prove that among $20$ consecutive positive integers there is an integer $d$ such that for every positive integer $n$ the following inequality holds
$$n \sqrt{d} \left\{n \sqrt {d} \right \} > \dfrac{5}{2}$$
where by $\left \{x \right \}$ denotes the fractional part of the real number $x$. The fractional part of the real number $x$ is defined as the difference between the largest integer that is less than or equal to $x$ to the actual number $x$.
[i](Serbia)[/i]
2023 ELMO Shortlist, N3
Let \(a\), \(b\), and \(n\) be positive integers. A lemonade stand owns \(n\) cups, all of which are initially empty. The lemonade stand has a [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]If at any moment, \(a\) completely empty cups are available, the filling machine spends the next \(a\) minutes filling those \(a\) cups simultaneously and doing nothing else. [*]If at any moment, \(b\) completely full cups are available, the emptying machine spends the next \(b\) minutes emptying those \(b\) cups simultaneously and doing nothing else. [/list] Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of \(a\) and \(b\), the least possible value of \(n\).
[i]Proposed by Raymond Feng[/i]
LMT Speed Rounds, 2016.20
Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$.
[i]Proposed by Nathan Ramesh
2007 Indonesia TST, 2
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
1979 IMO Shortlist, 5
Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$
1953 Moscow Mathematical Olympiad, 254
Given a $101\times 200$ sheet of graph paper, we start moving from a corner square in the direction of the square’s diagonal (not the sheet’s diagonal) to the border of the sheet, then change direction obeying the laws of light’s reflection. Will we ever reach a corner square?
[img]https://cdn.artofproblemsolving.com/attachments/b/8/4ec2f4583f406feda004c7fb4f11a424c9b9ae.png[/img]
2007 Nicolae Coculescu, 2
Let be two sequences $ \left( a_n \right)_{n\ge 0} , \left( b_n \right)_{n\ge 0} $ satisfying the following system:
$$ \left\{ \begin{matrix} a_0>0,& \quad a_{n+1} =a_ne^{-a_n} , &\quad\forall n\ge 0 \\ b_{0}\in (0,1) ,& \quad b_{n+1} =b_n\cos \sqrt{b_n} ,& \quad\forall n\ge 0 \end{matrix} \right. $$
Calculate $ \lim_{n\to\infty} \frac{a_n}{b_n} . $
[i]Florian Dumitrel[/i]
V Soros Olympiad 1998 - 99 (Russia), 8.5
Points $A$, $B$ and $C$ lie on one side of the angle with the vertex at point $O$, and points $A'$, $B'$ and $C'$ lie on the other. It is known that$ B$ is the midpoint of the segment $AC$, $B'$ is the midpoint of the segment $A'C'$, and lines $AA'$, $BB'$ and $CC'$ are parallel (fig.). Prove that the centers of the circles circumscribed around the triangles $OAC$, $OA'C$ and $OBB'$ lie on the same straight line.
[img]https://cdn.artofproblemsolving.com/attachments/d/6/92831077781bc45f25e9f71077034f84753a59.png[/img]
1999 Putnam, 5
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]
LMT Guts Rounds, 2015
[u]Round 9[/u]
[b]p25.[/b] For how many nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$ is the sum of the elements divisble by $32$?
[b]p26.[/b] America declared independence in $1776$. Take the sum of the cubes of the digits of $1776$ and let that equal $S_1$. Sum the cubes of the digits of $S_1$ to get $S_2$. Repeat this process $1776$ times. What is $S_{1776}$?
[b]p27.[/b] Every Golden Grahams box contains a randomly colored toy car, which is one of four colors. What is the expected number of boxes you have to buy in order to obtain one car of each color?
[u]Round 10[/u]
[b]p28.[/b] Let $B$ be the answer to Question $29$ and $C$ be the answer to Question $30$. What is the sum of the square roots of $B$ and $C$?
[b]p29.[/b] Let $A$ be the answer to Question $28$ and $C$ be the answer to Question $30$. What is the sum of the sums of the digits of $A$ and $C$?
[b]p30.[/b] Let $A$ be the answer to Question $28$ and $B$ be the answer to Question $29$. What is $A + B$?
[u]Round 11[/u]
[b]p31.[/b] If $x + \frac{1}{x} = 4$, find $x^6 + \frac{1}{x^6}$.
[b]p32.[/b] Given a positive integer $n$ and a prime $p$, there is are unique nonnegative integers $a$ and $b$ such
that $n = p^b \cdot a$ and $gcd (a, p) = 1$. Let $v_p(n)$ denote this uniquely determined $a$. Let $S$ denote the set of the first 20 primes. Find $\sum_{ p \in S} v_p \left(1 + \sum^{100}_{i=0} p^i \right)$.
[b]p33. [/b] Find the maximum value of n such that $n+ \sqrt{(n - 1) +\sqrt{(n - 2) + ... +\sqrt{1}}} < 49$
(Note: there would be $n - 1$ square roots and $n$ total terms).
[u]Round 12[/u]
[b]p34.[/b] Give two numbers $a$ and $b$ such that $2015^a < 2015! < 2015^b$. If you are incorrect you get
$-5$ points; if you do not answer you get $0$ points; otherwise you get $\max \{20-0.02(|b - a| - 1), 0\}$ points, rounded down to the nearest integer.
[b]p35.[/b] Twin primes are prime numbers whose difference is $2$. Let $(a, b)$ be the $91717$-th pair of twin primes, with $a < b$. Let $k = a^b$, and suppose that $j$ is the number of digits in the base $10$ representation of $k$. What is $j^5$? If the correct answer is $n$ and you say $m$, you will receive $\max \left(20 - | \log \left(| \frac{m}{n} |\right), 0 \right)$ points, rounded down to the nearest integer.
[b]p36.[/b] Write down any positive integer. Let the sum of the valid submissions (i.e. positive integer submissions) for all teams be $S$. One team will be chosen randomly, according to the following distribution:
if your team's submission is $n$, you will be chosen with probability $\frac{n}{S}$ . The amount of points that the chosen team will win is the greatest integer not exceeding $\min \{K, \frac{ 10000}{S} \}$. $K$ is a predetermined secret value.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2005 Finnish National High School Mathematics Competition, 4
The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer.
Show that permuting the order of digits one can obtain an integer divisible by $7.$
2021 China National Olympiad, 6
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$
1958 AMC 12/AHSME, 41
The roots of $ Ax^2 \plus{} Bx \plus{} C \equal{} 0$ are $ r$ and $ s$. For the roots of
\[ x^2 \plus{} px \plus{} q \equal{} 0
\]
to be $ r^2$ and $ s^2$, $ p$ must equal:
$ \textbf{(A)}\ \frac{B^2 \minus{} 4AC}{A^2}\qquad
\textbf{(B)}\ \frac{B^2 \minus{} 2AC}{A^2}\qquad
\textbf{(C)}\ \frac{2AC \minus{} B^2}{A^2}\qquad \\
\textbf{(D)}\ B^2 \minus{} 2C\qquad
\textbf{(E)}\ 2C \minus{} B^2$
2023 AMC 12/AHSME, 9
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
[asy]
size(200);
defaultpen(linewidth(0.6pt)+fontsize(10pt));
real y = sqrt(3);
pair A,B,C,D,E,F,G,H;
A = (0,0);
B = (0,y);
C = (y,y);
D = (y,0);
E = ((y + 1)/2,y);
F = (y, (y - 1)/2);
G = ((y - 1)/2, 0);
H = (0,(y + 1)/2);
fill(H--B--E--cycle, gray);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
[/asy]
$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$
2021 Peru PAGMO TST, P7
In a country there are $2021$ cities. Each pair of cities is either linked by a single road or not linked at all. It is known that for any subset of $2019$ cities, the total number of roads between them is the same. If the total number of roads in that country is $A$, find all possible values of $A$.
1996 Spain Mathematical Olympiad, 4
For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.
2020 BMT Fall, 1
Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2018 PUMaC Algebra A, 4
Suppose real numbers $a, b, c, d$ satisfy $a + b + c + d = 17$ and $ab + bc + cd + da = 46$. If the minimum possible value of $a^2 + b^2 + c^2 + d^2$ can be expressed as a rational number $\frac{p}{q}$ in simplest form, find $p + q$.
1965 All Russian Mathematical Olympiad, 062
What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?
1978 Canada National Olympiad, 6
Sketch the graph of $x^3 + xy + y^3 = 3$.
2000 Singapore Team Selection Test, 2
In a triangle $ABC$, $\angle C = 60^o$, $D, E, F$ are points on the sides $BC, AB, AC$ respectively, and $M$ is the intersection point of $AD$ and $BF$. Suppose that $CDEF$ is a rhombus. Prove that $DF^2 = DM \cdot DA$
1999 Gauss, 17
In a “Fibonacci” sequence of numbers, each term beginning with the third, is the sum of the previous two terms. The first number in such a sequence is 2 and the third is 9. What is the eighth term in the sequence?
$\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 107 \qquad \textbf{(D)}\ 152 \qquad \textbf{(E)}\ 245$