Found problems: 85335
A deck of $ n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $ (n\plus{}1)/2$.
Let $a,\ b>0$ be real numbers, $n\geq 2$ be integers.
Evaluate $I_n=\int_{-\infty}^{\infty} \frac{exp(ia(x-ib))}{(x-ib)^n}dx.$
Anana has an ordered $n$-tuple $(a_1,a_2,...,a_n)$ if integers. Banana may make a guess on Anana's ordered integer $n$-tuple $(x_1,x_2,...,x_n)$, upon which Anana will reveal the product of differences $(a_1-x_1)(a_2-x_2)...(a_n-x_n)$. How many guesses does Banana need to figure out Anana's $n$-tuple for certain?
Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that
[list]
[*] ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and
[*] $\angle BEN = \angle MED$.
[/list]
[i]Proposed by Evan Chen[/i]
There are given $111$ coins and a $n\times n$ table divided into unit cells. This coins are placed inside the unit cells (one unit cell may contain one coin, many coins, or may be empty), such that the difference between the number of coins from two neighbouring cells (that have a common edge) is $1$. Find the maximal $n$ for this to be possible.
Is there a quadratic trinomial $f(x)$ with integer coefficients such that $f(f(\sqrt{2}))=0$ ?
[i]A. Khrabrov[/i]
Show that the recursion $n=x_n(x_{n-1}+x_n+x_{n+1})$, $n=1,2,\ldots$, $x_0=0$ has exaclty one nonnegative solution.
(translated by L. Erdős)
Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$
Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$,
$P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.
Call a natural number $n$ faithful if there exist natural numbers $a<b<c$ such that $a|b,$ and $b|c$ and $n=a+b+c.$
$(i)$ Show that all but a finite number of natural numbers are faithful.
$(ii)$ Find the sum of all natural numbers which are not faithful.
Given is a triangle $ABC$. The points $P, Q$ lie on the extensions of $BC$ beyond $B, C$, respectively, such that $BP=BA$ and $CQ=CA$. Prove that the circumcenter of triangle $APQ$ lies on the angle bisector of $\angle BAC$.
Let $P$ units be the increase in the circumference of a circle resulting from an increase in $\pi$ units in the diameter. Then $P$ equals:
$\textbf{(A)}\ \dfrac{1}{\pi} \qquad
\textbf{(B)}\ \pi \qquad
\textbf{(C)}\ \dfrac{\pi^2}{2} \qquad
\textbf{(D)}\ \pi^2 \qquad
\textbf{(E)}\ 2\pi $
Alice is planning a trip from the Bay Area to one of $5$ possible destinations (each of which is serviced by only $1$ airport) and wants to book two flights, one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways may she plan her flight itinerary?
Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.
Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that
\[
\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.
\]
When does equality hold?
Solve the system of equations for variables $x,y$, where $\{a,b\}\in\mathbb{R}$ are constants and $a\neq 0$.
\[x^2 + xy = a^2 + ab\] \[y^2 + xy = a^2 - ab\]
Given quadrilateral $ABCD$. Its sidelines$ AB$ and $CD$ intersect in point $K$. It's diagonals intersect in point $L$. It is known that line $KL$ pass through the centroid of $ABCD$. Prove that $ABCD$ is trapezoid.
(F.Nilov)
Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
Consider the system of equations $$a + 2b + 3c + \ldots + 26z = 2020$$ $$b + 2c + 3d + \ldots + 26a = 2019$$ $$\vdots$$ $$y + 2z + 3a + \ldots + 26x = 1996$$ $$z + 2a + 3b + \ldots + 26y = 1995$$ where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of $$z + 2y + 3x + \dots + 26a.$$
[i]Proposed by Joshua Hsieh[/i]
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.
A circle with center $O$ is tangent to the sides of the angle with the vertex $A$ at the points B and C. Let M be a point on the larger of the two arcs $BC$ of this circle (different from $B$ and $C$) such that $M$ does not lie on the line $AO$. Lines $BM$ and $CM$ intersect the line $AO$ at the points $P$ and $Q$ respectively. Let $K$ be the foot of the perpendicular drawn from $P$ to $AC$ and $L$ be the foot of the perpendicular drawn from $Q$ to $AB$. Prove that the lines $OM$ and $KL$ are perpendicular.
$ AB \equal{} AC$, $ \angle BAC \equal{} 80^\circ$. Let $ E$ be a point inside $ \triangle ABC$ such that $ AE \equal{} EC$ and $ \angle EAC \equal{} 10^\circ$. What is the measure of $ \angle EBC$?
$\textbf{(A)}\ 10^\circ \qquad\textbf{(B)}\ 15^\circ \qquad\textbf{(C)}\ 20^\circ \qquad\textbf{(D)}\ 25^\circ \qquad\textbf{(E)}\ 30^\circ$
Let $M,N, P$ be the midpoints of the sides $BC,CA,AB$ of the triangle $ABC$, respectively, and let $G$ be the centroid of the triangle. Prove that if $BMGP$ is cyclic and $2BN = \sqrt3 AB$ , then triangle $ABC$ is equilateral.
A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.