Found problems: 85335
In triangle $\vartriangle ABC$, $E$ and $F$ are the feet of the altitudes from $B$ to $\overline{AC}$ and $C$ to $\overline{AB}$, respectively. Line $\overleftrightarrow{BC}$ and the line through $A$ tangent to the circumcircle of $ABC$ intersect at $X$. Let $Y$ be the intersection of line $\overleftrightarrow{EF}$ and the line through $A$ parallel to $\overline{BC}$. If $XB = 4$, $BC = 8$, and $EF = 4\sqrt3$, compute $XY$.
Decide if there is a permutation $a_1,a_2,\cdots,a_{6666}$ of the numbers $1,2,\cdots,6666$ with the property that the sum $k+a_k$ is a perfect square for all $k=1,2,\cdots,6666$
Two wheels with fixed hubs, each having a mass of $1 \text{ kg}$, start from rest, and forces are applied as shown. Assume the hubs and spokes are massless, so that the rotational inertia is $I = mR^2$. In order to impart identical angular accelerations about their respective hubs, how large must $F_2$ be?
[asy]
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw(circle((0,0),0.5));
draw((1, 0.5)--(0,0.5)--(0,-0.5),BeginArrow);
draw((-0.5,0)--(0.5,0));
draw((-0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(0.5*sqrt(2)/2,-0.5*sqrt(2)/2));
draw((0.5*sqrt(2)/2, 0.5*sqrt(2)/2)--(-0.5*sqrt(2)/2,-0.5*sqrt(2)/2));
label("$R$ = 0.5 m", (0, -0.5),S);
label("$F_1$ = 1 N",(1,0.5),N);
draw(circle((3,0.5),1));
draw((4.5,1.5)--(3,1.5)--(3,-0.5),BeginArrow);
draw((2,0.5)--(4,0.5));
draw((3-sqrt(2)/2, 0.5+sqrt(2)/2)--(3+sqrt(2)/2, 0.5-sqrt(2)/2));
draw((3+sqrt(2)/2, 0.5+sqrt(2)/2)--(3-sqrt(2)/2,0.5-sqrt(2)/2));
label("$F_2$", (4.5, 1.5), N);
label("$R$ = 1 m",(3, -0.5),S);
[/asy]
$ \textbf{(A)}\ 0.25\text{ N}\qquad\textbf{(B)}\ 0.5\text{ N}\qquad\textbf{(C)}\ 1\text{ N}\qquad\textbf{(D)}\ 2\text{ N}\qquad\textbf{(E)}\ 4\text{ N} $
Determine the highest possible value of:
$$S = a_1a_2a_3 + a_4a_5a_6 +... + a_{2017}a_{2018}a_{2019} + a_{2020}$$
where $(a_1, a_2, a_3,..., a_{2020})$ is a permutation of $(1,2,3,..., 2020)$.
Clarification: In $S$, each term, except the last one, is the multiplication of three numbers.
Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$.
Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$.
[i]AMM Magazine[/i]
In the plane, what is the locus of points of the sum of the squares of whose distances from $n$ fixed points is a constant? What restrictions, stated in geometric terms, must be put on the constant so that the locus is non-null?
Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal.
Find all natural numbers $n$ for which both $n^n + 1$ and $(2n)^{2n} + 1$ are prime numbers.
[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance?
[b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.
A certain value of $x$ satisfies $1 + x + 5 - 1 = 7$. What is this value of $x$?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{impossible to determine}$
We play a game. The pot starts at $\$0$. On every turn, you flip a fair coin. If you flip heads, I add $\$100$ to the pot. If you flip tails, I take all of the money out of the pot, and you are assessed a "strike". You can stop the game before any flip and collect the contents of the pot, but if you get 3 strikes, the game is over and you win nothing. Find, with proof, the expected value of your winnings if you follow an optimal strategy.
Find all polynomials $P(x)$ with integer coefficients such that, for all integers $a$ and $b$, $P(a+b) - P(b)$ is a multiple of $P(a)$.
Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE \equal{} \angle ABC$.
In a school with 5 different grades there are 250 girls and 250 boys. Each grade has the same number of students. for a competition of knowledge wants to form teams of a female and a male who are of the same grade. If in each grade there are at least $19$ females and $19$ males. Find the greatest amount of teams that can be formed.
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?
$ \textbf{(A)}\ \frac{1}{36} \qquad
\textbf{(B)}\ \frac{1}{24} \qquad
\textbf{(C)}\ \frac{1}{18} \qquad
\textbf{(D)}\ \frac{1}{12} \qquad
\textbf{(E)}\ \frac{1}{9}$
As shown in figure , points $D,E,F$ lies the sides $AB$, $BC$ , $CA$ of the acute angle $\vartriangle ABC$ respectively. If $\angle EDC = \angle CDF$, $\angle FEA=\angle AED$, $\angle DFB =\angle BFE$, prove that the $CD$, $AE$, $BF$ are the altitudes of $\vartriangle ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/5ddf48e298ad1b75691c13935102b26abe73c1.png[/img]
A function f : $R$ $\rightarrow$ $R$ satisfies the property $f(x^2) - f^2(x) \geq 1/4$ for all x.
Verify if the function is one-one.
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.
[b]p1.[/b] Ten math students take a test, and the average score on the test is $28$. If five students had an average of $15$, what was the average of the other five students' scores?
[b]p2.[/b] If $a\otimes b = a^2 + b^2 + 2ab$, find $(-5\otimes 7) \otimes 4$.
[b]p3.[/b] Below is a $3 \times 4$ grid. Fill each square with either $1$, $2$ or $3$. No two squares that share an edge can have the same number. After filling the grid, what is the $4$-digit number formed by the bottom row?
[img]https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png[/img]
[b]p4.[/b] What is the angle in degrees between the hour hand and the minute hand when the time is $6:30$?
[b]p5.[/b] In a small town, there are some cars, tricycles, and spaceships. (Cars have $4$ wheels, tricycles have $3$ wheels, and spaceships have $6$ wheels.) Among the vehicles, there are $24$ total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town?
[b]p6.[/b] You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails?
[b]p7.[/b] In the below diagram, $\angle ABC$ and $\angle BCD$ are right angles. If $\overline{AB} = 9$, $\overline{BD} = 13$, and $\overline{CD} = 5$, calculate $\overline{AC}$.
[img]https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png[/img]
[b]p8.[/b] Out of $100$ customers at a market, $80$ purchased oranges, $60$ purchased apples, and $70$ purchased bananas. What is the least possible number of customers who bought all three items?
[b]p9.[/b] Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following:
Francis: Fred ate the cookies.
Fred: Ted did not eat the cookies.
Ted: Francis is lying.
If exactly one of them is telling the truth, who ate all the cookies?
[b]p11.[/b] Let $ABC$ be a triangle with a right angle at $A$. Suppose $\overline{AB} = 6$ and $\overline{AC} = 8$. If $AD$ is the perpendicular from $A$ to $BC$, what is the length of $AD$?
[b]p12.[/b] How many three digit even numbers are there with an even number of even digits?
[b]p13.[/b] Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line.
[b]p14.[/b] A man $5$ foot, $10$ inches tall casts a $14$ foot shadow. $20$ feet behind the man, a flagpole casts ashadow that has a $9$ foot overlap with the man's shadow. How tall (in inches) is the flagpole?
[b]p15.[/b] Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has $n$ balls and 3 divides $n$, then he throws away a third of the balls. If $3$ does not divide $n$ but $2$ divides $n$, then he throws away half of them. If neither $3$ nor $2$ divides $n$, he stops throwing away the balls. If he began with $1458$ balls, after how many steps does he stop throwing away balls?
[b]p16.[/b] Oski has $50$ coins that total to a value of $82$ cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter).
[b]p17.[/b] Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC$. Suppose $\overline{MA} = \overline{MB} = \overline{MC} = 2$ and $\angle ACB = 30^o$. Find the area of the triangle.
[b]p18.[/b] A spirited integer is a positive number representable in the form $20^n + 13k$ for some positive integer $n$ and any integer $k$. Determine how many spirited integers are less than $2013$.
[b]p19. [/b]Circles of radii $20$ and $13$ are externally tangent at $T$. The common external tangent touches the circles at $A$, and $B$, respectively where $A \ne B$. The common internal tangent of the circles at $T$ intersects segment $AB$ at $X$. Find the length of $AX$.
[b]p20.[/b] A finite set of distinct, nonnegative integers $\{a_1, ... , a_k\}$ is called admissible if the integer function $f(n) = (n + a_1) ... (n + a_k)$ has no common divisor over all terms; that is, $gcd \left(f(1), f(2),... f(n)\right) = 1$ for any integer$ n$. How many admissible sets only have members of value less than $10$? $\{4\}$ and $\{0, 2, 6\}$ are such sets, but $\{4, 9\}$ and $\{1, 3, 5\}$ are not.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
4. A three-dimensional rectangular box with dimensions $X$, $Y$, and $Z$ has faces whose surface areas are $24$, $24$, $48$, $48$, $72$, and $72$ square units. What is $X + Y + Z$?
$\textbf{(A)} \text{ 18} \qquad \textbf{(B)} \text{ 22} \qquad \textbf{(C)} \text{ 24} \qquad \textbf{(D)} \text{ 30} \qquad \textbf{(E)} \text{ 36}$
The largest prime factor of $199^4 + 4$ has four digits. Compute the second largest prime factor.
Compute the number of integers $1 \leq n \leq 1024$ such that the sequence $\lceil n \rceil$, $\lceil n/2 \rceil$, $\lceil n/4 \rceil$, $\lceil n/8 \rceil$, $\ldots$ does not contain any multiple of $5$.
[i]Proposed by Sean Li[/i]
Dan has a fair $6$-sided die with faces labeled $1,2,3,4,+,$ and $-.$ In order to complete the equation
\[ \underline{\qquad} \ \underline{\qquad} \ \underline{\qquad}=\underline{\qquad}, \]
Dan repeatedly rolls his die and fills in a blank with the character he obtained, starting with the leftmost blank and progressing rightward. The probability that, when all blanks are filled, Dan forms a true equation, is $\frac{p}{q},$ where $p$ and $q$ are relatively prime integers. Find $p+q.$
A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.
The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$.
Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.