Found problems: 85335
2011 Today's Calculation Of Integral, 723
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$
2023 Chile TST Ibero., 1
Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers
\[
a_n = 4^n c + \frac{4^n - (-1)^n}{5}
\]
contains at least one perfect square.
2006 Tournament of Towns, 6
Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)
2005 Estonia Team Selection Test, 6
Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$.
Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.
2008 iTest Tournament of Champions, 2
Let $A$ be the number of $12$-digit words that can be formed by from the alphabet $\{0,1,2,3,4,5,6\}$ if each pair of neighboring digits must differ by exactly $1$. Find the remainder when $A$ is divided by $2008$.
1985 IMO Longlists, 13
Find the average of the quantity
\[(a_1 - a_2)^2 + (a_2 - a_3)^2 +\cdots + (a_{n-1} -a_n)^2\]
taken over all permutations $(a_1, a_2, \dots , a_n)$ of $(1, 2, \dots , n).$
2008 Princeton University Math Competition, B2
What is $3(2 \log_4 (2(2 \log_3 9)))$ ?
2011 USAJMO, 3
For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_1), l(P_2), l(P_3)$ form an equilateral triangle $\triangle$. Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.
1997 Federal Competition For Advanced Students, Part 2, 2
We define the following operation which will be applied to a row of bars being situated side-by-side on positions $1, 2, \ldots ,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by- side in such a way that all bars form a new row and are situated on positions $1, \ldots,M$. From an initial number $a_0 > 0$ of bars there originates a sequence $(a_n)_{n \geq 0}$, where an is the number of bars after having applied the operation $n$ times.
[b](a)[/b] Prove that for no $n > 0$ can we have $a_n = 1997$.
[b](b)[/b] Determine all natural numbers that can only occur as $a_0$ or $a_1$.
1974 IMO Shortlist, 4
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
2006 China Team Selection Test, 3
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:
(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$
(2) $d \mid (x_1+x_2+ \cdots x_n)$
Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
2010 AMC 10, 4
A book that is to be recorded onto compact discs takes $ 412$ minutes to read aloud. Each disc can hold up to $ 56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
$ \textbf{(A)}\ 50.2 \qquad
\textbf{(B)}\ 51.5 \qquad
\textbf{(C)}\ 52.4 \qquad
\textbf{(D)}\ 53.8 \qquad
\textbf{(E)}\ 55.2$
1994 IMC, 6
Find
$$\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}$$
2016 ASDAN Math Tournament, 9
A cyclic quadrilateral $ABCD$ has side lengths $AB=14$, $BC=19$, $CD=26$, and $DA=29$. Compute the sine of the smaller angle between diagonals $AC$ and $BD$.
2005 China Team Selection Test, 2
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?
2022 JHMT HS, 7
Let $a$ be the unique real number $x$ satisfying $xe^x = 2$. Find a closed-form expression for
\[ \int_{a}^{\infty} \frac{x + 1}{x\sqrt{(xe^x)^{11} - 1}}\,dx. \]
You may express your answer in terms of elementary operations, functions, and constants.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P5
There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a [i]friendly triplet[/i] if any two students in the group are friends. Determine the maximal possible number of friendly triplets.
[i]Proposed by Nikola Velov[/i]
1998 South africa National Olympiad, 6
You are given $n$ squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?
2017 Costa Rica - Final Round, F1
Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.
2011 Sharygin Geometry Olympiad, 25
Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?
2007 Estonia Math Open Senior Contests, 7
Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of
some $ n$ consecutive integers is a perfect square?
1979 USAMO, 5
A certain organization has $n$ members, and it has $n\plus{}1$ three-member committees, no two of which have identical member-ship. Prove that there are two committees which share exactly one member.
Mid-Michigan MO, Grades 7-9, 2006
[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following:
$\begin{tabular}{ccccc}
& a & b & c & a \\
+ & & d & d & e \\
& & & d & e \\
\hline
d & f & f & d & d \\
\end{tabular}$
[b]p2.[/b] Explain whether it possible that the sum of two squares of positive whole numbers has all digits equal to $1$:
$$n^2 + m^2 = 111...111$$
[b]p3. [/b]Two players play the following game on an $8 \times 8$ chessboard. The first player can put a rook on an arbitrary square. Then the second player can put another rook on a free square that is not controlled by the first rook. Then the first player can put a new rook on a free square that is not controlled by the rooks on the board. Then the second player can do the same, etc. A player who cannot put a new rook on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Show that the difference $9^{2008} - 7^{2008}$ is divisible by $10$.
[b]p5.[/b] Is it possible to find distict positive whole numbers $a, b, c, d, e$ such that
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1?$$
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 Kyiv City MO Round 1, Problem 2
$ABCD$ is a trapezoid with $BC\parallel AD$ and $BC = 2AD$. Point $M$ is chosen on the side $CD$ such that $AB = AM$. Prove that $BM \perp CD$.
[i]Proposed by Bogdan Rublov[/i]
2016 Taiwan TST Round 3, 2
Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions:
$(i)$ $a_0=a_n=1$;
$(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$;
$(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
$(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$.
Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.