Found problems: 85335
The perimeter of the triangle $ABC$ is equal to $2p$, the length of the side$ AC$ is equal to $b$, the angle $ABC$ is equal to $\beta$. A circle with center at point $O$, inscribed in this triangle, touches the side $BC$ at point $K$. Calculate the area of the triangle $BOK$.
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.
Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages $ 40$ miles per hour, he arrives at his workplace three minutes late. When he averages $ 60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?
$ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 58$
Find all pairs of integers $ \left(m,n\right) $ such that $ \left(m,n+1\right) = 1 $ and $$ \sum\limits_{k=1}^{n}{\frac{m^{k+1}}{k+1}\binom{n}{k}} \in \mathbb{N} $$
Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.
Determine all pairs $(n,m)$ of positive integers for which there exists an infinite sequence $\{x_k\}$ of $0$'s and $1$'s with the properties that if $x_i=0$ then $x_{i+m}=1$ and if $x_i = 1$ then $x_{i+n} = 0.$
A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Prove that the lines $PI, A_1C_1$, and $B_1D_1$ are concurrent.
Ankan Bhattacharya, USA
Two convex polygons can be placed into a square with the side $1$ without intersection. Prove that at least one polygon has the perimeter that is less than or equal to $3,5$ .
Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and
\[f(x)f'(y)=f'(x)f(y).\]
In a group of $kn$ persons, each person knows more than $(k-1)n$ others ($k,n$ are positive integers). Prove that one can choose $k+1$ persons from this group so that each chosen person knows all the others chosen.
Note: If a person $A$ knows $B$, then $B$ knows $A$.
Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.
Find all positive integer solutions $(x,y,z,n)$ of equation $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$, where $n\ge 2$ and $z \le 5\times 2^{2n}$.
Find the value of
\[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]
A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals:
${{ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15}\qquad\textbf{(E)}\ 18} $
Let $I$ be the incenter of the acute triangle $\triangle ABC$, and $BI$, $CI$ intersect the altitude of $\triangle ABC$ through $A$ at $U$, $V$, respectively. The circle with $AI$ as a diameter intersects $\odot(ABC)$ again at $T$, and $\odot(TUV)$ intersects the segment $BC$ and $\odot(ABC)$ at $P$, $Q$, respectively. Let $R$ be another intersection of $PQ$ and $\odot(ABC)$. Show that $AR\parallel BC$.
A positive integer $x$ is $k$-[i]equivocal [/i] if there exists two positive integers $b$, $b'$ such that when $x$ is represented in base $b$ and base $b'$, the two representations have digit sequences of length $k$ that are permutations of each other. The smallest $2$-equivocal number is $7$, since $7$ is $21$ in base $3$ and $12$ in base $5$. Find the smallest $3$-equivocal number.
For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)\pmod{2}$, where $c(n,i) \in \left\{ {0,1} \right\}$. Define
\[f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}\]
where $m,n,q$ are positive integers and $q + 1 \ne {2^\alpha }$ for any $\alpha \in \mathbb N$. Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.
How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$?
$ \textbf{(A) \ }198\qquad \textbf{(B) \ } 199 \qquad \textbf{(C) \ } 398 \qquad \textbf{(D) \ }399\qquad \textbf{(E) \ } 597$
The circle $\omega$ with center at point $I$ inscribed in an triangle $ABC$ ($AB\neq AC$) touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The circumcircles of triangles $ABC$ and $AEF$ intersect secondary at point $K.$ The lines $EF$ and $AK$ intersect at point $X$ and intersects the line $BC$ at points $Y$ and $Z$, respectively. The tangent lines to $\omega$, other than $BC$, passing through points $Y$ and $Z$ touch $\omega$ at points $P$ and $Q$, respectively. Let the lines $AP$ and $KQ$ intersect at the point $R$. Prove that if $M$ is a midpoint of segment $YZ,$ then $IR\perp XM$.
Prove that if
$$\sqrt{x + a} +\sqrt{y+b}+\sqrt{z + c} =\sqrt{y + a} +\sqrt{z + b} +\sqrt{x + c} =\sqrt{z + a} +\sqrt{x+b}+\sqrt{y+c}$$
for some $a, b, c, x, y, z$, then $x = y = z$ or $a = b = c$.
Find all functions $f : R \to R$ such that for all real $x,y$, $f(f(x)+y) = f(x^2 -y)+4y f(x)$
Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
What is the sum of the roots of $z^{12} = 64$ that have a positive real part?
$\textbf{(A) }2 \qquad\textbf{(B) }4 \qquad\textbf{(C) }\sqrt{2} +2\sqrt{3}\qquad\textbf{(D) }2\sqrt{2}+ \sqrt{6} \qquad\textbf{(E) }(1 + \sqrt{3}) + (1+\sqrt{3})i$
On blackboard there are $10$ different positive integers which sum is equal to $62$. Prove that product of those numbers is divisible with $60$