Found problems: 85335
2014 SEEMOUS, Problem 4
a) Prove that $\lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2$.
b) Find the limit $\lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right)$.
2016 BAMO, 3
The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$.
Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$).
Show that, for every integer $k$ greater than or equal to $2$,
[list=i]
[*] $A$ and $B$ have the same set of distinct prime factors.
[*] $A+1$ and $B+1$ have the same set of distinct prime factors.
[/list]
Estonia Open Senior - geometry, 1994.2.2
The two sides $BC$ and $CD$ of an inscribed quadrangle $ABCD$ are of equal length. Prove that the area of this quadrangle is equal to $S =\frac12 \cdot AC^2 \cdot \sin \angle A$
2017 Junior Regional Olympiad - FBH, 3
On blackboard there are $10$ different positive integers which sum is equal to $62$. Prove that product of those numbers is divisible with $60$
2014-2015 SDML (High School), 2
The number $15$ is written on a blackboard. A move consists of erasing the number $x$ and replacing it with $x+y$ where $y$ is a randomly chosen number between $1$ and $5$ (inclusive). The game ends when the number on the blackboard exceeds $51$. Which number is most likely to be on the blackboard at the end of the game?
$\text{(A) }52\qquad\text{(B) }53\qquad\text{(C) }54\qquad\text{(D) }55\qquad\text{(E) }56$
1998 Irish Math Olympiad, 3
Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$.
2015 Danube Mathematical Competition, 3
Determine all positive integers $n$ such that all positive integers less than or equal to $n$ and relatively prime to $n$ are pairwise coprime.
2005 IMC, 2
Let $f: \mathbb{R}\to\mathbb{R}$ be a function such that $(f(x))^{n}$ is a polynomial for every integer $n\geq 2$. Is $f$ also a polynomial?
2008 Saint Petersburg Mathematical Olympiad, 2
Point $O$ is the center of the circle into which quadrilateral $ABCD$ is inscribed. If angles $AOC$ and $BAD$ are both equal to $110$ degrees and angle $ABC$ is greater than angle $ADC$, prove that $AB+AD>CD$.
Fresh translation.
2022 Irish Math Olympiad, 8
8. The Equation [i]AB[/i] X [i]CD[/i] = [i]EFGH[/i], where each of the letters [i]A[/i], [i]B[/i], [i]C[/i], [i]D[/i], [i]E[/i], [i]F[/i], [i]G[/i], [i]H[/i] represents a different digit and the values of [i]A[/i], [i]C[/i] and [i]E[/i] are all nonzero, has many solutions, e.g., 46 X 85 =3910. Find the smallest value of the four-digit number [i]EFGH[/i] for which there is a solution.
2010 Princeton University Math Competition, 2
Find the largest positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$.
2016 Macedonia National Olympiad, Problem 2
A magic square is a square with side 3 consisting of 9 unit squares, such that the numbers written in the unit squares (one number in each square) satisfy the following property: the sum of the numbers in each row is equal to the sum of the numbers in each column and is equal to the sum of all the numbers written in any of the two diagonals.
A rectangle with sides $m\ge3$ and $n\ge3$ consists of $mn$ unit squares. If in each of those unit squares exactly one number is written, such that any square with side $3$ is a magic square, then find the number of most different numbers that can be written in that rectangle.
2016 239 Open Mathematical Olympiad, 8
Given a natural number $k>1$. Find the smallest number $\alpha$ satisfying the following condition. Suppose that the table $(2k + 1) \times (2k + 1)$ is filled with real numbers not exceeding $1$ in absolute value, and the sums of the numbers in all lines are equal to zero. Then you can rearrange the numbers so that each number remains in its row and all the sums over the columns will be at most $\alpha$.
2022 BMT, 8
Oliver is at a carnival. He is offered to play a game where he rolls a fair dice and receives $\$1$ if his roll is a $1$ or $2$, receives $\$2$ if his roll is a $3$ or $4$, and receives $\$3$ if his roll is a $5$ or $6$. Oliver plays the game repeatedly until he has received a total of at least $\$2$. What is the probability that he ends with $\$3$?
2011 Today's Calculation Of Integral, 697
Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$
2013 Finnish National High School Mathematics Competition, 1
The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.
2021 Bangladeshi National Mathematical Olympiad, 12
Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where both Gamakichi And Gamatatsu land on?
2008 China Northern MO, 5
Assume $n$ is a positive integer and integer $a$ is the root of the equation $$x^4+3ax^2+2ax-2\times 3^n=0.$$ Find all $n$ and $ a$ that satisfy the conditions.
2005 Turkey Team Selection Test, 3
We are given 5040 balls in k different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest k such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no two balls of the same color are in two neighboring bags.
Kyiv City MO Juniors 2003+ geometry, 2003.8.5
Three segments $2$ cm, $5$ cm and $12$ cm long are constructed on the plane. Construct a trapezoid with bases of $2$ cm and $5$ cm, the sum of the sides of which is $12$ cm, and one of the angles is $60^o$.
(Bogdan Rublev)
2008 Junior Balkan Team Selection Tests - Moldova, 4
The square table $ 10\times 10$ is divided in squares $ 1\times1$. In each square $ 1\times1$ is written one of the numers $ \{1,2,3,...,9,10\}$. Numbers from any two adjacent or diagonally adjacent squares are reciprocal prime. Prove, that there exists a number, which is written in this table at least 17 times.
1959 AMC 12/AHSME, 5
The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is:
$ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 256.25\qquad\textbf{(E)}\ -16$
2023 India National Olympiad, 1
Let $S$ be a finite set of positive integers. Assume that there are precisely 2023 ordered pairs $(x,y)$ in $S\times S$ so that the product $xy$ is a perfect square. Prove that one can find at least four distinct elements in $S$ so that none of their pairwise products is a perfect square.
[i]Note:[/i] As an example, if $S=\{1,2,4\}$, there are exactly five such ordered pairs: $(1,1)$, $(1,4)$, $(2,2)$, $(4,1)$, and $(4,4)$.
[i]Proposed by Sutanay Bhattacharya[/i]
2013 Harvard-MIT Mathematics Tournament, 4
Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers whose sum is $20$. Determine with proof the smallest possible value of \[ \displaystyle\sum_{1\le i \le j \le 5} \lfloor a_i + a_j \rfloor. \]
1984 Tournament Of Towns, (077) 2
A set of numbers $a_1, a_2 , . . . , a_{100}$ is obtained by rearranging the numbers $1 , 2,..., 100$ . Form the numbers
$b_1=a_1$
$b_2= a_1 + a_2$
$b_3=a_1 + a_2 + a_3$
...
$b_{100}=a_1 + a_2 + ...+a_{100}$
Prove that among the remainders on dividing the numbers by $100 , 11$ of them are different .
( L . D . Kurlyandchik , Leningrad)