Found problems: 85335
2011 Today's Calculation Of Integral, 694
Prove the following inequality:
\[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\]
created by kunny
2000 National Olympiad First Round, 10
$N$ is a $50-$digit number (in the decimal scale). All digits except the $26^{\text{th}}$ digit (from the left) are $1$. If $N$ is divisible by $13$, what is the $26^{\text{th}}$ digit?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{More information is needed}
$
2015 India IMO Training Camp, 1
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
1988 China Team Selection Test, 4
Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$.
(i) Find $r_5$.
(ii) Find $r_7$.
(iii) Find $r_k$ for $k \in \mathbb{N}$.
1972 IMO Longlists, 38
Congruent rectangles with sides $m(cm)$ and $n(cm)$ are given ($m, n$ positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)
1990 Putnam, A3
Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to $ \dfrac {5}{2} $.
2021 Estonia Team Selection Test, 1
a) There are $2n$ rays marked in a plane, with $n$ being a natural number. Given that no two marked rays have the same direction and no two marked rays have a common initial point, prove that there exists a line that passes through none of the initial points of the marked rays and intersects with exactly $n$ marked rays.
(b) Would the claim still hold if the assumption that no two marked rays have a common initial point was dropped?
2008 AMC 10, 21
Ten chairs are evenly spaced around a round table and numbered clockwise from $ 1$ through $ 10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?
$ \textbf{(A)}\ 240\qquad
\textbf{(B)}\ 360\qquad
\textbf{(C)}\ 480\qquad
\textbf{(D)}\ 540\qquad
\textbf{(E)}\ 720$
2010 Hanoi Open Mathematics Competitions, 8
If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, fi nd $n$.
2018 Yasinsky Geometry Olympiad, 4
In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1, \angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.
2024 Junior Macedonian Mathematical Olympiad, 4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.
[i]Proposed by Ilija Jovčeski[/i]
1966 AMC 12/AHSME, 39
In base $R_1$ the expanded fraction $F_1$ becomes $0.373737...$, and the expanded fraction $F_2$ becomes $0.737373...$. In base $R_2$ fraction $F_1$, when expanded, becomes $0.252525...$, while fraction $F_2$ becomes $0.525252...$. The sum of $R_1$ and $R_2$, each written in base ten is:
$\text{(A)}\ 24 \qquad
\text{(B)}\ 22\qquad
\text{(C)}\ 21\qquad
\text{(D)}\ 20\qquad
\text{(E)}\ 19$
1958 AMC 12/AHSME, 26
A set of $ n$ numbers has the sum $ s$. Each number of the set is increased by $ 20$, then multiplied by $ 5$, and then decreased by $ 20$. The sum of the numbers in the new set thus obtained is:
$ \textbf{(A)}\ s \plus{} 20n\qquad
\textbf{(B)}\ 5s \plus{} 80n\qquad
\textbf{(C)}\ s\qquad
\textbf{(D)}\ 5s\qquad
\textbf{(E)}\ 5s \plus{} 4n$
2005 German National Olympiad, 6
The sequence $x_0,x_1,x_2,.....$ of real numbers is called with period $p$,with $p$ being a natural number, when for each $p\ge2$, $x_n=x_{n+p}$.
Prove that,for each $p\ge2$ there exists a sequence such that $p$ is its least period
and $x_{n+1}=x_n-\frac{1}{x_n}$ $(n=0,1,....)$
2015 All-Russian Olympiad, 7
A scalene triangle $ABC$ is inscribed within circle $\omega$. The tangent to the circle at point $C$ intersects line $AB$ at point $D$. Let $I$ be the center of the circle inscribed within $\triangle ABC$. Lines $AI$ and $BI$ intersect the bisector of $\angle CDB$ in points $Q$ and $P$, respectively. Let $M$ be the midpoint of $QP$. Prove that $MI$ passes through the middle of arc $ACB$ of circle $\omega$.
2022 Indonesia MO, 4
Given a regular $26$-gon. Prove that for any $9$ vertices of that regular $26$-gon, then there exists three vertices that forms an isosceles triangle.
2013 Baltic Way, 20
Find all polynomials $f$ with non-negative integer coefficients such that for all primes $p$ and positive integers $n$ there exist a prime $q$ and a positive integer $m$ such that $f(p^n)=q^m$.
2003 IMC, 5
a) Show that for each function $f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}$, there exists a function $g:\mathbb{Q}\rightarrow \mathbb{R}$ with $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{Q}$.
b) Find a function $f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, for which there is no function $g:\mathbb{Q}\rightarrow \mathbb{R}$ such that $f(x,y) \leq g(x)+g(y) $ for all $x,y\in \mathbb{R}$.
2021 Dutch IMO TST, 1
Let $\Gamma$ be the circumscribed circle of a triangle $ABC$ and let $D$ be a point at line segment $BC$. The circle passing through $B$ and $D$ tangent to $\Gamma$ and the circle passing through $C $and $D$ tangent to $\Gamma$ intersect at a point $E \ne D$. The line $DE$ intersects $\Gamma$ at two points $X$ and $Y$ . Prove that $|EX| = |EY|$.
2001 Tournament Of Towns, 6
Several numbers are written in a row. In each move, Robert chooses any two adjacent numbers in which the one on the left is greater than the one on the right, doubles each of them and then switches them around. Prove that Robert can make only a finite number of moves.
2024 SEEMOUS, P2
Let $A,B\in\mathcal{M}_n(\mathbb{R})$ two real, symmetric matrices with nonnegative eigenvalues. Prove that $A^3+B^3=(A+B)^3$ if and only if $AB=O_n$.
1983 IMO Longlists, 20
Let $f$ and $g$ be functions from the set $A$ to the same set $A$. We define $f$ to be a functional $n$-th root of $g$ ($n$ is a positive integer) if $f^n(x) = g(x)$, where $f^n(x) = f^{n-1}(f(x)).$
(a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$-th functional roots for each positive integer $n.$
(b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$
2023 MOAA, 4
Andy has $4$ coins $c_1, c_2, c_3, c_4$ such that the probability that coin $c_i$ with $1 \leq i \leq 4$ lands tails is $\frac{1}{2^i}$. Andy flips each coin exactly once. The probability that only one coin lands on heads can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Anthony Yang[/i]
2019 Miklós Schweitzer, 6
Let $d$ be a positive integer and $1 < a \le (d+2)/(d+1)$. For given $x_0, x_1,\dots, x_d \in (0, a-1)$, let $x_{k+1} = x_k (a - x_{k-d})$, $k \ge d$. Prove that $\lim_{k \to \infty} x_k = a-1$.
1994 Balkan MO, 4
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
[i]Bulgaria[/i]