Found problems: 85335
Let $ f$ be a continuous function on the unit interval $ [0,1]$. Show that \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)\] and \[ \lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).\]
Let $k\geq 4$ be an integer number. $P(x)\in\mathbb{Z}[x]$ such that $0\leq P(c)\leq k$ for all $c=0,1,...,k+1$. Prove that $P(0)=P(1)=...=P(k+1)$.
The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.
In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.
Find all finite sets $M \subset R, |M| \ge 2$, satisfying the following condition:
[i]for all $a, b \in M, a \ne b$, the number $a^3 - \frac{4}{9}b$ also belongs to $M$.
[/i]
(I. Voronovich)
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.
Find all non negative integers $a{}$ and $b{}$ satisfying $5^a+6=31^b$
[i]Proposed by Erik Füredi, Budapest[/i]
Let $a$ and $b$ positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satis es $2 \left(\frac{a}{b}\right) = m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions:
$(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$
$(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$
[i]Proposed by Netherlands.[/i]
Find all the three-digit numbers such that it equals to the arithmetic mean of the six numbers obtained by rearranging its digits.
Let $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$. Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that:
$$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2) $$
Suppose function $f:[0,\infty)\to[0,\infty)$ satisfies
(1)$\forall x,y \geq 0,$ we have $f(x)f(y)\leq y^2f(\frac{x}{2})+x^2f(\frac{y}{2})$;
(2)$\forall 0 \leq x \leq 1, f(x) \leq 2016$.
Prove that $f(x)\leq x^2$ for all $x\geq 0$.
For any $k\in\mathbb{Z}^+$ , prove that:-
$2(\sqrt{k+1}-\sqrt{k})<\frac{1}{\sqrt{k}}<2(\sqrt{k}-\sqrt{k-1})$
Also compute integral part of $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{10000}}$.
The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.
Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$
Let $ABCD$ be a square of side paper $10$ and $P$ a point on side $BC$. By folding the paper along the $AP$ line, point $B$ determines the point $Q$, as seen in the figure. The line $PQ$ cuts the side $CD$ at $R$. Calculate the perimeter of the triangle $ PCR$
[img]https://3.bp.blogspot.com/-ZSyCUznwutE/XNY7cz7reQI/AAAAAAAAKLc/XqgQnjm8DQYq6Q7fmCAKJwKt3ihoL8AuQCK4BGAYYCw/s400/may%2B2013%2Bl1.png[/img]
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
Let $K$ be an arbitrary point on side $BC$ of triangle $ABC$, and $KN$ be a bisector of triangle $AKC$. Lines $BN$ and $AK$ meet at point $F$, and lines $CF$ and $AB$ meet at point $D$. Prove that $KD$ is a bisector of triangle $AKB$.
What is the sum of distinct real numbers $x$ such that $(2x^2+5x+9)^2=56(x^3+1)$?
$
\textbf{(A)}\ 3
\qquad\textbf{(B)}\ \dfrac{7}{4}
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \dfrac{9}{2}
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
$(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) = $
$\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$
Let $ABC$ be a triangle with integer sides in which $AB<AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB,AC)>1$.
The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because one can see the whole area of the museum standing at either point. Is it true that from any point of the $AB$ section one can see the whole museum?
The positive integers $ x$ and $ y$ are the two smallest positive integers for which the product of $ 360$ and $ x$ is a square and the product of $ 360$ and $ y$ is a cube. What is the sum of $ x$ and $ y$?
$ \textbf{(A)}\ 80 \qquad
\textbf{(B)}\ 85 \qquad
\textbf{(C)}\ 115 \qquad
\textbf{(D)}\ 165 \qquad
\textbf{(E)}\ 610$
The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if:
$ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$