Found problems: 85335
A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$, point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$?
For any three nonnegative reals $a$, $b$, $c$, prove that
$\left|ca-ab\right|+\left|ab-bc\right|+\left|bc-ca\right|\leq\left|b^{2}-c^{2}\right|+\left|c^{2}-a^{2}\right|+\left|a^{2}-b^{2}\right|$.
[i]Generalization.[/i] For any $n$ nonnegative reals $a_{1}$, $a_{2}$, ..., $a_{n}$, prove that
$\sum_{i=1}^{n}\left|a_{i-1}a_{i}-a_{i}a_{i+1}\right|\leq\sum_{i=1}^{n}\left|a_{i}^{2}-a_{i+1}^{2}\right|$.
Here, the indices are cyclic modulo $n$; this means that we set $a_{0}=a_{n}$ and $a_{n+1}=a_{1}$.
darij
In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.
Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $.0001$ of the correct answer will be accepted.
Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .
A function $f$, defined on the set of real numbers $\mathbb{R}$ is said to have a [i]horizontal chord[/i] of length $a>0$ if there is a real number $x$ such that $f(a+x)=f(x)$. Show that the cubic $$f(x)=x^3-x\quad \quad \quad \quad (x\in \mathbb{R})$$ has a horizontal chord of length $a$ if, and only if, $0<a\le 2$.
Consider the isosceles right triangle $ABC$ with $\angle BAC = 90^\circ$. Let $\ell$ be the line passing through $B$ and the midpoint of side $AC$. Let $\Gamma$ be the circumference with diameter $AB$. The line $\ell$ and the circumference $\Gamma$ meet at point $P$, different from $B$. Show that the circumference passing through $A,\ C$ and $P$ is tangent to line $BC$ at $C$.
Define a string to be doubly palindromic if it can be split into two (non-empty) parts that are read the same both backwards and forwards. For example hannahhuh is doubly palindromic as it can be split into hannah and huh. How many doubly palindromic strings of length 9 using only the letters $\{a, b, c, d\}$ are there?
Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.
Let $f(n,k)$ with $n,k\in\mathbb Z_{\geq 2}$ be defined such that $\frac{(kn)!}{(n!)^{f(n,k)}}\in\mathbb Z$ and $\frac{(kn)!}{(n!)^{f(n,k)+1}}\not\in\mathbb Z$
Define $m(k)$ such that for all $k$, $n\geq m(k)\implies f(n,k)=k$. Show that $m(k)$ exists and furthermore that $m(k)\leq \mathcal{O}\left(k^2\right)$
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$
Let $\vartriangle ABC$ have side lengths $AB = 4,BC = 6,CA = 5$. Let $M$ be the midpoint of $BC$ and let $P$ be the point on the circumcircle of $\vartriangle ABC$ such that $\angle MPA = 90^o$. Let $D$ be the foot of the altitude from $B$ to $AC$, and let $E$ be the foot of the altitude from $C$ to $AB$. Let $PD$ and $PE$ intersect line $BC$ at $X$ and $Y$ , respectively. Compute the square of the area of $\vartriangle AXY$ .
Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.
$ G $ is the centroid of $ ABC. $ The incircle of $ ABC $ touches $ BC,CA,AB $ at $ D,E,F, $ respectively. Show that $ ABC $ is equilateral if and only if $ BC\cdot\overrightarrow{GD}+ AC\cdot\overrightarrow{GE} +AB\cdot\overrightarrow{GF} =0. $
[i]Marian Ursărescu[/i]
Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$.
When does equality holds?
(Karl Czakler)
The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_{n}$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n\geq 0$, $d$ divides $F_{n+2020}-F_n$. Compute the remainder when $d$ is divided by $1001$.
[i]Proposed by Ankit Bisain[/i]
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?
$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$
Aladdin has a set of coins with weights $ 1, 2, \ldots, 20$ grams. He can ask Genie about any two coins from the set which one is heavier, but he should pay Genie some other coin from the set before. (So, with every question the set of coins becomes smaller.) Can Aladdin find two coins from the set with total weight at least $ 28$ grams?
We define a [i]chessboard polygon[/i] to be a polygon whose sides are situated along lines of the form $ x \equal{} a$ or $ y \equal{} b$, where $ a$ and $ b$ are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles. Finally, a [i]tasteful tiling[/i] is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $ 3 \times 4$ rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner.
[asy]size(300); pathpen = linewidth(2.5);
void chessboard(int a, int b, pair P){
for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j)
if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6));
D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle);
}
chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12));
/* draw lines */
D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3));[/asy] a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s).
Find all possible initial position of the black stone, to make all stones black by finite actions.
A point $C$ is marked on a chord $AB$ of a circle $\omega.$ Let $D$ be the midpoint of $AC,$ and $O$ be the center of the circle $\omega.$ The circumcircle of the triangle $BOD$ intersects the circle $\omega$ again at point $E$ and the straight line $OC$ again at point $F.$ Prove that the circumcircle of the triangle $CEF$ touches $AB.$
For any continuous real-valued function $f$ defined on the interval $[0,1],$ let \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1],$ then \[\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.\]
In the figure of [url]http://www.artofproblemsolving.com/Forum/download/file.php?id=50643&mode=view[/url]
$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$.
The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$,
and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$.
If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals:
$ \textbf{(A)}\ k^6\qquad
\textbf{(B)}\ 5k^3\qquad
\textbf{(C)}\ k^3\qquad
\textbf{(D)}\ 243\qquad
\textbf{(E)}\ 125$
Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.