Found problems: 85335
The circle inscribed in the triangle $ABC$ with center $I$ touches the side $BC$ at the point $D$. The line $DI$ intersects the side $AC$ at the point $M$. The tangent from $M$ to the inscribed circle, different from $AC$, intersects the side $AB$ at the point $N$. The line $NI$ intersects the side $BC$ at the point $P$. Prove that $AB = BP$.
For some positive integer $n$, a coin will be flipped $n$ times to obtain a sequence of $n$ heads and tails. For each flip of the coin, there is probability $p$ of obtaining a head and probability $1-p$ of obtaining a tail, where $0<p<1$ is a rational number.
Kim writes all $2^n$ possible sequences of $n$ heads and tails in two columns, with some sequences in the left column and the remaining sequences in the right column. Kim would like the sequence produced by the coin flips to appear in the left column with probability $1/2$.
Determine all pairs $(n,p)$ for which this is possible.
Show that there exist infinitely many positive integers $n$ such that $n^{2}+1$ divides $n!$.
Let $ABCD$ be a rectangle and the arbitrary points $E\in (CD)$ and $F \in (AD)$. The perpendicular from point $E$ on the line $FB$ intersects the line $BC$ at point $P$ and the perpendicular from point $F$ on the line $EB$ intersects the line $AB$ at point $Q$. Prove that the points $P, D$ and $Q$ are collinear.
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.
(a) Prove that $f_n$ is well defined.
(b) Find all polynomials $f \in \mathbb Z[X]$ such that for all $n \in \mathbb N$, $n \geq 2$, the function $f_n$ is surjective.
[i]Bogdan Enescu[/i]
Find all real number $x$ which could be represented as
$x = \frac{a_0}{a_1a_2 . . . a_n} + \frac{a_1}{a_2a_3 . . . a_n} + \frac{a_2}{a_3a_4 . . . a_n} + . . . + \frac{a_{n-2}}{a_{n-1}a_n} + \frac{a_{n-1}}{a_n}$ , with $n, a_1, a_2, . . . . , a_n$ are positive integers and $1 = a_0 \leq a_1 < a_2 < . . . < a_n$
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $. Find the sum of the sum of the coordinates of all such points.
Let $p > 1$ be a natural number. Consider the set $F_p$ of all non-constant sequences of non-negative integers that satisfy the recursive relation $a_{n+1} = (p+1)a_n - pa_{n-1}$ for all $n > 0$.
Show that there exists a sequence ($a_n$) in $F_p$ with the property that for every other sequence ($b_n$) in $F_p$, the inequality $a_n \le b_n$ holds for all $n$.
Given a circle $\omega$ of radius $r$ and a point $A$, which is far from the center of the circle at a distance $d<r$. Find the geometric locus of vertices $C$ of all possible $ABCD$ rectangles, where points $B$ and $D$ lie on the circle $\omega$.
$n$ people live in a town, and they are members of some clubs (residents can be members of more than one club). No matter how we choose some (but at least one) clubs, there is a resident of the town who is the member of an odd number of the chosen clubs. Prove that the number of clubs is at most $n$.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
A ballot box contains the votes for the election of two candidates $A$ and $B$. It is known that candidate $A$ has $6$ votes and candidate $B$ has $9$. Find the probability that, when carrying out the scrutiny, candidate $B$ always goes first.
Let $A$ be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that $A$ has two elements at most.
For positive integer numbers $a$, $b$ and $c$ it is known that $a^2+b^2+c^2$ and $a^3+b^3+c^3$ are both divisible by $a+b+c$. In addition, $gcd(a+b+c, 6) = 1$. Prove that $a^5+b^5+c^5$ is divisible by $(a+b+c)^2$.
[i] A. Antropov [/i]
Let $ K $ be the midpoint of the side $ AB $ of a given triangle $ ABC $. Let $ L $ and $ M$ be points on the sides $ AC $ and $ BC$, respectively, such that $ \angle CLK = \angle KMC $. Prove that the perpendiculars to the sides $ AB, AC, $ and $ BC $ passing through $ K,L, $ and $M$, respectively, are concurrent.
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel?
(The cogwheels may be rotated about their common axis, but not turned over.)
(b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]
Let $p, q, r, s, t \in \mathbb{R}^{*}_{+}$ satisfying:
i) $p^2 + pq + q^2 = s^2$
ii) $q^2 + qr + r^2 = t^2$
iii) $r^2 + rp + p^2 = s^2 - st + t^2$
Prove that
\[\frac{s^2 - st + t^2}{s^2t^2} = \frac{r^2}{q^2t^2} + \frac{p^2}{q^2s^2} - \frac{pr}{q^2ts}\]
Suppose $a,b$ and $c$ are real numbers that satisfy $a+b+c=5$ and $\tfrac{1}{a}+\tfrac{1}{b}+\tfrac{1}{c}=\tfrac15$. Find the greatest possible value of $a^3+b^3+c^3$.
Five rays $\overrightarrow{OA}$,$\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$, and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\angle EOD = 2\angle COB$, $\angle COB = 2\angle BOA$, while $\angle DOC = 3\angle BOA$. If $E$, $O$, $A$ are collinear with $O$ between $A$ and $E$, what is the degree measure of $\angle DOB?$
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$.
Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.
Let $n$ be a positive integer. Find all real numbers $x_1,x_2 ,..., x_n$ such that $$\prod_{k=1}^{n}(x_k^2+ (k + 2)x_k + k^2 + k + 1) =\left(\frac{3}{4}\right)^n (n!)^2$$
Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]
Let $\Omega$ be a circle of radius $8$ centered at point $O$, and let $M$ be a point on $\Omega$. Let $S$ be the set of points $P$ such that $P$ is contained within $\Omega$ , or such that there exists some rectangle $ABCD$ containing $P$ whose center is on $\Omega$ with$ AB = 4$, $BC = 5$, and $BC \parallel OM$. Find the area of $S$.