Found problems: 85335
2001 Romania National Olympiad, 4
Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$.
a) Show that:
\[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\]
b) Show that:
\[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]
2016 SGMO, Q1
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for any pair of naturals $m,n$,
$$\gcd(f(m),n) = \gcd(m,f(n)).$$
2014 Dutch IMO TST, 1
Determine all pairs $(a,b)$ of positive integers satisfying
\[a^2+b\mid a^2b+a\quad\text{and}\quad b^2-a\mid ab^2+b.\]
1999 Gauss, 3
Susan wants to place 35.5 kg of sugar in small bags. If each bag holds 0.5 kg, how many bags are needed?
$\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 53 \qquad \textbf{(D)}\ 70 \qquad \textbf{(E)}\ 71$
2006 MOP Homework, 2
Points $P$ and $Q$ lies inside triangle $ABC$ such that $\angle ACP =\angle BCQ$ and $\angle CAP = \angle BAQ$. Denote by $D,E$, and $F$ the feet of perpendiculars from $P$ to lines $BC,CA$, and $AB$, respectively. Prove that if $\angle DEF = 90^o$, then $Q$ is the orthocenter of triangle $BDF$.
1996 All-Russian Olympiad Regional Round, 10.8
There are $1996$ points marked on a straight line at regular intervals. Petya colors half of them red and the rest blue. Then Vasya divides them into pairs ''red'' - ''blue'' so that the sum distances between points in pairs was maximum. Prove that this maximum does not depend on what coloring Petya made.
2012 Online Math Open Problems, 36
Let $s_n$ be the number of solutions to $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$, where $a_1,a_2,a_3$ and $a_4$ are elements of the set $\{2, 3, 5, 7\}$ and $b_1$ and $b_2$ are elements of the set $\{ 1, 2, 3, 4\}$. Find the number of $n$ for which $s_n$ is odd.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]$s_n$ is the number of [i]ordered[/i] solutions $(a_1, a_2, a_3, a_4, b_1, b_2)$ to the equation, where each $a_i$ lies in $\{2, 3, 5, 7\}$ and each $b_i$ lies in $\{1, 2, 3, 4\}$. [/hide]
2001 All-Russian Olympiad, 2
The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.
2016 IMAR Test, 2
Given a positive integer $n$, does there exist a planar polygon and a point in its plane such that every line through that point meets the boundary of the polygon at exactly $2n$ points?
1992 Irish Math Olympiad, 2
If $a_1$ is a positive integer, form the sequence $a_1,a_2,a_3,\dots$ by letting $a_2$ be the product of the digits of $a_1$, etc.. If $a_k$ consists of a single digit, for some $k\ge 1$, $a_k$ is called a [i]digital root[/i] of $a_1$. It is easy to check that every positive integer has a unique root. $($For example, if $a_1=24378$, then $a_2=1344$, $a_3=48$, $a_4=32$, $a_5=6$, and thus $6$ is the digital root of $24378.)$ Prove that the digital root of a positive integer $n$ equals $1$ if, and only if, all the digits of $n$ equal $1$.
2014 Estonia Team Selection Test, 1
In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.
2014 India Regional Mathematical Olympiad, 1
Three positive real numbers $a,b,c$ are such that $a^2+5b^2+4c^2-4ab-4bc=0$. Can $a,b,c$ be the lengths of te sides of a triangle? Justify your answer.
2004 German National Olympiad, 3
Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.
2021 SAFEST Olympiad, 2
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
1990 IMO Longlists, 65
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.
2006 All-Russian Olympiad Regional Round, 11.7
Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.
2008 Indonesia TST, 3
Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$.
(i) Prove that $CK$ is the angle bisector of $\angle ACB$.
(ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.
2014 Postal Coaching, 5
Determine all polynomials $f$ with integer coefficients with the property that for any two distinct primes $p$ and $q$, $f(p)$ and $f(q)$ are relatively prime.
2022/2023 Tournament of Towns, P3
Let us call a positive integer [i]pedestrian[/i] if all its decimal digits are equal to 0 or 1. Suppose that the product of some two pedestrian integers also is pedestrian. Is it necessary in this case that the sum of digits of the product equals the product of the sums of digits of the factors?
[i]Viktor Kleptsyn, Konstantin Knop[/i]
2004 Canada National Olympiad, 5
Let $ T$ be the set of all positive integer divisors of $ 2004^{100}$. What is the largest possible number of elements of a subset $ S$ of $ T$ such that no element in $ S$ divides any other element in $ S$?
1949-56 Chisinau City MO, 47
Determine the type of triangle if the lengths of its sides $a, b, c$ satisfy the relation $$a^4 + b^4 + c^4 = a^2b^2 + b^2c^2 + c^2a^2$$
2024 AMC 10, 3
What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?
$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
2000 Tournament Of Towns, 4
In how many ways can $31$ squares be marked on an $8 \times 8$ chessboard so that no two of the marked squares have a common side?
(R Zhenodarov)
2011 Balkan MO, 2
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
2003 China Western Mathematical Olympiad, 4
$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.