Found problems: 85335
In triangle $ABC$, let $S$ be on $BC$ and $T$ be on $AC$ so that $AS \perp BC$ and $BT \perp AC$, and let $AS$ and $BT$ intersect at $H$. Let $O$ be the center of the circumcircle of $\vartriangle AHT, P$ be the center of the circumcircle of $\vartriangle BHS$, and $G$ be the other point of intersection (besides $H$) of the two circles. Let $GH$ and $OP$ intersect at $X$. If $AB = 14, BH = 6$, and HA = 11, then $XO - XP$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
Square $\mathcal S$ has vertices $(1,0)$, $(0,1)$, $(-1,0)$ and $(0,-1)$. Points $P$ and $Q$ are independently selected, uniformly at random, from the perimeter of $\mathcal S$. Determine, with proof, the probability that the slope of line $PQ$ is positive.
[i]Proposed by Isabella Grabski[/i]
Let $k,r \in \mathbb N$ and let $x\in (0,1)$ be a rational number given in decimal representation \[ x = 0.a_1a_2a_3a_4 \ldots . \] Show that if the decimals $a_k, a_{k+r}, a_{k+2r}, \ldots$ are canceled, the new number obtained is still rational.
[i]Dan Schwarz[/i]
Let $c_0,\,c_1,\,c_2,\,\ldots$ be a sequence defined so that
\[
\frac{1-3x-\sqrt{1-14x+9x^2}}{4}=\sum_{k=0}^\infty c_kx^k
\]
for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\,\ldots,\,n\}$. Find the determinant of $A$.
How many letters in the word UNCOPYRIGHTABLE have at least one line of symmetry?
Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1\equal{}1$, $ a_2\equal{}\dfrac{4}{3}$, and \[ a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2}\] and \[ 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.\]
[i]Fajar Yuliawan, Bandung[/i]
Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.
The equation $ x^6 \minus{} 3x^5 \minus{} 6x^3 \minus{} x \plus{} 8$ has
$ \textbf{(A)}\ \text{no real roots} \qquad$
$ \textbf{(B)}\ \text{exactly two distinct negative roots} \qquad$
$ \textbf{(C)}\ \text{exactly one negative root} \qquad$
$ \textbf{(D)}\ \text{no negative roots, but at least one positive root} \qquad$
$ \textbf{(E)}\ \text{none of these}$
Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?
Show that any positive rational number can be represented as the sum of three positive rational cubes.
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$ at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.
Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$.Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$
Find all prime numbers of the form $10101...01$.
Let $\mathcal{C}$ be the set of all twice differentiable functions $f:[0,1] \to \mathbb{R}$ with at least two (not necessarily distinct) zeros and $|f''(x)| \le 1,$ for all $x \in [0,1].$ Find the greatest value of the integral $$\int\limits_0^1 |f(x)| \mathrm{d}x$$ when $f$ runs through the set $\mathcal{C},$ as well as the functions that achieve this maximum.
[i]Note: A differentiable function $f$ has two zeros in the same point $a$ if $f(a)=f'(a)=0.$[/i]
Find the ordered quadruple of digits $(A,B,C,D)$ with $A > B > C > D$, such that
$$\begin{tabular}{ccccc}
& A & B & C & D \\
- & D & C & B & A \\
\hline
= & B & D & A & C \\
\end{tabular}$$
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.
There are $ n \geq 3$ job openings at a factory, ranked $1$ to $ n$ in order of increasing pay. There are $ n$ job applicants, ranked from $1$ to $ n$ in order of increasing ability. Applicant $ i$ is qualified for job $ j$ if and only if $ i \geq j.$ The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified AND which is lower in rank than any job already filled. (Under these rules, job $1$ is always filled, and hiring terminates thereafter.) Show that applicants $ n$ and $ n \minus{} 1$ have the same probability of being hired.
How many multiples of $17$ are there between $23$ and $227$?
In an in finite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows:
[list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$
- and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list]
Prove that in this sequence
[b](a) [/b] the number $1$ occurs infi nitely many times;
[b](b)[/b] each positive integer occurs infi nitely many times.
(The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )
[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus?
[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins?
[b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits?
$\begin{tabular}{ccccc}
& & & 9 & 5 \\
x & & & * & * \\
\hline
& & & * & * \\
+ & 1 & * & * & \\
\hline
& * & * & * & * \\
\end{tabular}$
Find all solutions.
[b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes.
[b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
For an integer $k$ let $T_k$ denote the number of $k$-tuples of integers $(x_1,x_2,...x_k)$ with $0\le x_i < 73$ for each $i$, such that $73|x_1^2+x_2^2+...+x_k^2-1$. Compute the remainder when $T_1+T_2+...+T_{2017}$ is divided by $2017$.
[i]Proposed by Vincent Huang
Find the maximum of
\[P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\]
where $x,y,z$ are positive real numbers.
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers?
In a triangle $ABC$, the angle bisector at $A,B,C$ meet the opposite sides at $A_1,B_1,C_1$, respectively. Prove that if the quadrilateral $BA_1B_1C_1$ is cyclic, then
$$\frac{AC}{AB+BC}=\frac{AB}{AC+CB}+\frac{BC}{BA+AC}.$$