Found problems: 85335
Find the sum of all $3$-digit numbers whose digits, when read from left to right, form a strictly increasing sequence. (Numbers with a leading zero, e.g. ”$087$” or ”$002$”, are not counted as having $3$ digits.)
Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$.
[i]Proposed by Evan Chen[/i]
In the film there is $n$ roles. For each $i$ ($1 \le i \le n$), the role of number $i$ can play $a_i$ a person, and one person can play only one role. Every day a casting is held, in which participate people for $n$ roles, and from each role only one person. Let $p$ be a prime number such that $p \ge a_1, \ldots, a_n, n$. Prove that you can have $p^k$ castings such that if we take any $k$ people who are tried in different roles, they together participated in some casting ($k$ is a natural number not exceeding $n$ ).
Determine all functions $f: \mathbb{R} \backslash \{0 \} \rightarrow \mathbb{R}$ such that, for all nonzero $x$:
$$ f(\frac{1}{x}) \ge 1 -f(x) \ge x^2f(x) $$
Pentagon $ABCDE$ is inscribed in circle $\Omega$. Line $AD$ meets $CE$ at $F$, and $P (\neq E, F)$ is a point on segment $EF$. The circumcircle of triangle $AFP$ meets $\Omega$ at $Q(\neq A)$ and $AC$ at $R(\neq A)$. Line $AD$ meets $BQ$ at $S$, and the circumcircle of triangle $DES$ meets line $BQ, BD$ at $T(\neq S), U(\neq D)$, respectively. Prove that if $F, P, T, S$ are concyclic, then $P, T, R, U$ are concyclic.
Let $x, y, z$ be three real distinct numbers such that
$$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.
Let $G$ be the centroid of the triangle $ABC$ in which the angle at $C$ is obtuse and $AD$ and $CF$ be the medians from $A$ and $C$ respectively onto the sides $BC$ and $AB$. If the points $\ B,\ D, \ G$ and $\ F$ are concyclic, show that $\dfrac{AC}{BC} \geq \sqrt{2}$. If further $P$ is a point on the line $BG$ extended such that $AGCP$ is a parallelogram, show that triangle $ABC$ and $GAP$ are similar.
Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that
\[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\]
Prove that there exists a point $ M$ in the plane of the pentagon such that
\[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\]
Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.
Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.
given $p_1,p_2,...$ be a sequence of integer and $p_1=2$,
for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $
prove that all primes appear in the sequence
(Proposed by Beatmania)
Given that $(1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ}) = 2^n$, find $n$.
Find all positive real numbers $c$ such that there are in finitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer.
(Slovakia)
Let $n$ be a five digit number (whose first digit is non-zero) and let $m$ be the four digit number formed from $n$ by removing its middle digit. Determine all $n$ such that $n/m$ is an integer.
Let $a_1,a_2,\dotsc$ be a sequence of real numbers from the interval $[0,1]$. Prove that there is a sequence $1\leqslant n_1<n_2<\dotsc$ of positive integers such that
$$A=\lim_{\substack{i,j\to \infty \\ i\neq j}} a_{n_i+n_j}$$exists, i.e., for every real number $\epsilon >0$, there is a constant $N_{\epsilon}$ that $|a_{n_i+n_j}-A|<\epsilon$ is satisfied for any pair of distinct indices $i,j>N_{\epsilon}$.
The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.
Let $ABC$ be a triangle with incenter $I$ and let $\Gamma$ be its circumcircle. Let $M$ be the midpoint of $BC$, $K$ the midpoint of the arc $BC$ which does not contain $A$, $L$ the midpoint of the arc $BC$ which contains $A$ and $J$ the reflection of $I$ by the line $KL$. The line $LJ$ intersects $\Gamma$ again at the point $T\neq L$. The line $TM$ intersects $\Gamma$ again at the point $S\neq T$. Prove that $S, I, M, K$ lie on the same circle.
Consider a triangle $\triangle ABC$ and some point $X$ on $BC$. The perpendicular from $X$ to $AB$ intersects the circumcircle of $\triangle AXC$ at $P$ and the perpendicular from $X$ to $AC$ intersects the circumcircle of $\triangle AXB$ at $Q$. Show that the line $PQ$ does not depend on the choice of $X$.
[i](Shining Sun, USA)[/i]
Let $a$, $b$, $c$ be positive real numbers. Prove that
\[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]
The sequence $a_1, a_2, ...$ of natural numbers satisfies $GCD(a_i, a_j)=GCD(i, j)$ for all $i \neq j$. Prove that $a_i=i$ for all $i$.
Prove that if $A{}$ and $B{}$ are $n\times n$ matrices with complex entries which satisfy \[A=AB-BA+A^2B-2ABA+BA^2+A^2BA-ABA^2,\]then $\det(A)=0$.
Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that:
\[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]
Let C be the number of ways to arrange the letters of the word CATALYSIS, T be the number of ways to arrange the letters of the word TRANSPORT, S be the number of ways to arrange the letters of the word STRUCTURE, and M be the number of ways to arrange the letters of the word MOTION. What is $\frac{C - T + S}{M}$ ?
Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.
[i]Proposed by Ankan Bhattacharya[/i]
In a triangle $ABC$ the incenter, the $B$-excenter and the $C$-excenter are $I, K$ and $L$, respectively. The perpendiculars at $B$ and $C$ to $BC$ intersect the lines $AC$ and $AB$ at $E$ and $F$, respectively. Prove that the circumcircles of $AEF, FIL, EIK$ concur.
$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a
$\textbf{(A)}~\text{Rectangle}$
$\textbf{(B)}~\text{Rhombus}$
$\textbf{(C)}~\text{Isosceles Trapezium}$
$\textbf{(D)}~\text{Square}$