Found problems: 85335
Determine the number of pairs of positive integers $x,y$ such that $x\le y$, $\gcd (x,y)=5!$ and $\text{lcm}(x,y)=50!$.
Let $f_1,f_2$ be nonnegative and concave functions. Then prove that
$$(f_1f_2)^{\frac{2^n-1}{n\cdot2^n}}\left(\frac{\displaystyle\prod_{k=1}^n\left(\sqrt[2^k]{f_1}+\sqrt[2^k]{f_2}\right)}{f_1+f_2}\right)^{\frac1n}$$
is concave.
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
Let $ABC$ and $BCD$ be equilateral triangles, such that $AB=1,$ and $A \neq D.$ Find the area of triangle $ABD.$
Let $n \ge 3$ be an integer. Find the minimal value of the real number $k_n$ such that for all positive numbers $x_1, x_2, ..., x_n$ with product $1$, we have $$\frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1.$$
Let $ a,b,c$ be three pairwise distinct real numbers such that $ a\plus{}b\plus{}c\equal{}6\equal{}ab\plus{}bc\plus{}ca\minus{}3$. Prove that $ 0<abc<4$.
Solve in natural the equation
$9^x-3^x=y^4+2y^3+y^2+2y$
_____________________________
Azerbaijan Land of the Fire :lol:
In acute-angled triangle $ABC$ with $|BC| < |BA|$, point $N$ is the midpoint of $AC$. The circle with diameter $AB$ intersects the bisector of $\angle B$ in two points: $B$ and $X$. Prove that $XN$ is parallel to $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/f0ae8f5df8f2cc1bb80de1ee1807dc845a87b3.png[/img]
In acute-angled triangle $ABC,$ $AB>AC.$ Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. The line passing through $H$ and parallel to $AB$ intersects line $AC$ at $M,$ and the line passing through $H$ and parallel to $AC$ intersects line $AB$ at $N.$ $L$ is the reflection of the point $H$ in $MN.$ Line $OL$ and $AH$ intersect at $K.$ Prove that $K,M,L,N$ are concyclic.
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.
Given an integer $n \ge 2$, solve in real numbers the system of equations \begin{align*}
\max\{1, x_1\} &= x_2 \\
\max\{2, x_2\} &= 2x_3 \\
&\cdots \\
\max\{n, x_n\} &= nx_1. \\
\end{align*}
$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$
There are $2$ runners on the perimeter of a regular hexagon, initially located at adjacent vertices. Every second, each of the runners independently moves either one vertex to the left, with probability $\frac{1}{2}$, or one vertex to the right, also with probability $\frac{1}{2}$. Find the probability that after a $2013$ second run (in which runners switch vertices $2013$ times each), the runners end up at adjacent vertices once again.
Let $H$ be the orthocenter of an acute triangle $ABC$. (The orthocenter is the point at the intersection of the three altitudes. An acute triangle has all angles less than $90^o$.) Draw three circles: one passing through $A, B$, and $H$, another passing through $B, C$, and $H$, and finally, one passing through $C, A$, and $H$. Prove that the triangle whose vertices are the centers of those three circles is congruent to triangle $ABC$.
[u]Round 1[/u]
[b]p1.[/b] What is $2 + 22 + 1 + 3 - 31 - 3$?
[b]p2.[/b] Let $ABCD$ be a rhombus. Given $AB = 5$, $AC = 8$, and $BD = 6$, what is the perimeter of the rhombus?
[b]p3.[/b] There are $2$ hats on a table. The first hat has $3$ red marbles and 1 blue marble. The second hat has $2$ red marbles and $4$ blue marbles. Jordan picks one of the hats randomly, and then randomly chooses a marble from that hat. What is the probability that she chooses a blue marble?
[u]Round 2[/u]
[b]p4.[/b] There are twelve students seated around a circular table. Each of them has a slip of paper that they may choose to pass to either their clockwise or counterclockwise neighbor. After each person has transferred their slip of paper once, the teacher observes that no two students exchanged papers. In how many ways could the students have transferred their slips of paper?
[b]p5.[/b] Chad wants to test David's mathematical ability by having him perform a series of arithmetic operations at lightning-speed. He starts with the number of cubic centimeters of silicon in his 3D printer, which is $109$. He has David perform all of the following operations in series each second:
$\bullet$ Double the number
$\bullet$ Subtract $4$ from the number
$\bullet$ Divide the number by $4$
$\bullet$ Subtract $5$ from the number
$\bullet$ Double the number
$\bullet$ Subtract $4$ from the number
Chad instructs David to shout out after three seconds the result of three rounds of calculations. However, David computes too slowly and fails to give an answer in three seconds. What number should David have said to Chad?
[b]p6.[/b] Points $D, E$, and $F$ lie on sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively, such that the following length conditions are true: $CD = AE = BF = 2$ and $BD = CE = AF = 4$. What is the area of triangle $ABC$?
[u]Round 3[/u]
[b]p7.[/b] In the $2, 3, 5, 7$ game, players count the positive integers, starting with $1$ and increasing, which do not contain the digits $2, 3, 5$, and $7$, and also are not divisible by the numbers $2, 3, 5$, and $7$. What is the fifth number counted?
[b]p8.[/b] If A is a real number for which $19 \cdot A = \frac{2014!}{1! \cdot 2! \cdot 2013!}$ , what is $A$?
Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ...\cdot 2 \cdot 1$.
[b]p9.[/b] What is the smallest number that can be written as both $x^3 + y^2$ and $z^3 + w^2$ for positive integers $x, y, z,$ and $w$ with $x \ne z$?
[u]Round 4[/u]
[i]Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. In addition, it is given that the answer to each of the following problems is a positive integer less than or equal to the problem number.
[/i]
[b]p10.[/b] Let $B$ be the answer to problem $11$ and let $C$ be the answer to problem $12$. What is the sum of a side length of a square with perimeter $B$ and a side length of a square with area $C$?
[b]p11.[/b] Let $A$ be the answer to problem $10$ and let $C$ be the answer to problem $12$. What is $(C - 1)(A + 1) - (C + 1)(A - 1)$?
[b]p12.[/b] Let $A$ be the answer to problem $10$ and let $B$ be the answer to problem $11$. Let $x$ denote the positive difference between $A$ and $B$. What is the sum of the digits of the positive integer $9x$?
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2915810p26040675]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Circles $k_1$ and $k_2$ intersect at $P$ and $Q$, and $A$ and $B$ are the tangency points of their common tangent that is closer to $P$ (where $A$ is on $k_1$ and $B$ on $k_2$). The tangent to $k_1$ at $P$ intersects $k_2$ again at $C$. The lines $AP$ and $BC$ meet at $R$. Show that the lines $BP$ and $BC$ are tangent to the circumcircle of triangle $PQR$.
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11:4$ . To the nearest whole percent, what percent of its games did the team lose?
$ \text{(A)}\ 24\qquad\text{(B)}\ 27\qquad\text{(C)}\ 36\qquad\text{(D)}\ 45\qquad\text{(E)}\ 73 $
In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.
Let $F_1=F_2=1$. We define inductively $F_{n+1}=F_n+F_{n-1}$ for all $n\geq 2$. Then the sum $$F_1+F_2+\cdots+F_{2017}$$ is
[list=1]
[*] Even but not divisible by 3
[*] Odd but divisible by 3
[*] Odd and leaves remainder 1 when divisible by 3
[*] None of these
[/list]
For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$?
[i]Proposed by Michael Tang[/i]
What is the smallest $r$ such that three disks of radius $r$ can completely cover up a unit disk?
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $CE$?
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, G, H;
A = (0,0);
B = (5,0);
C = (5,1.5);
D = (0,1.5);
E = (1,1.5);
F = (8,1.5);
G = (8,3.5);
H = (1,3.5);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
dot("A", A, SW);
dot("B", B, SE);
dot("C", C, SE);
dot("D", D, NW);
dot("E", E, NW);
dot("F", F, SE);
dot("G", G, NE);
dot("H", H, NW);
[/asy]
There are $n$ cells with indices from $1$ to $n$. Originally, in each cell, there is a card with the corresponding index on it. Vasya shifts the card such that in the $i$-th cell is now a card with the number $a_i$. Petya can swap any two cards with the numbers $x$ and $y$, but he must pay $2|x-y|$ coins. Show that Petya can return all the cards to their original position, not paying more than $|a_1-1|+|a_2-2|+\ldots +|a_n-n|$ coins.
Calculate the sum of digit of the number
$$9 \times 99 \times 9999 \times ... \times \underbrace{ 99...99}_{2^n}$$
where the number of nines doubles in each factor.
In a movie theatre there are $6$ VIP chairs labelled from $1$ to $6$. We call a few consecutive vacant chairs a block. In the online VIP seat reservation process the reservation of a seat consists of two steps: in the first step we choose the block, in the second step we reserve the first, last or middle seat (in case of a block of size even this means the middle chair with the smaller number) of that block. (In the second step the online system offers the three possibilities even though they might mean the same seat.) Benedek reserved all seats at some screeining. In how many ways could he do it if we distinguish two reservation if there were a step when Benedek chose a different option?
For instance, if the seats $ 1$ and $6$ are reserved, then there are two blocks, the first one consists of the seat $ 1$, the second block consists of the seats $3, 4$ and $5$. Two reservation orders are different if there is a chair that was reserved in a different step, or there is a chair that was reserved with different option (first, last or middle). So if there were $2$ VIP chairs, then the answer would have been $9$.
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum.
Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)