Found problems: 85335
1995 Irish Math Olympiad, 1
Prove that for every positive integer $ n$,
$ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$
2016 CCA Math Bonanza, L4.1
Determine the remainder when $$2^6\cdot3^{10}\cdot5^{12}-75^4\left(26^2-1\right)^2+3^{10}-50^6+5^{12}$$ is divided by $1001$.
[i]2016 CCA Math Bonanza Lightning #4.1[/i]
2007 Bulgarian Autumn Math Competition, Problem 10.4
Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.
2022 Thailand TST, 1
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
1988 Tournament Of Towns, (172) 5
Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point?
( N . Vasiliev)
2008 Germany Team Selection Test, 2
Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:
[b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color,
and
[b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$.
[i]Author: Gerhard Wöginger, Netherlands[/i]
1990 Tournament Of Towns, (272) 6
A deck consists of $n$ different cards. A move consists of taking out a group of cards in sequence from some place in the deck, and putting it back someplace else without changing the order within the group or turning any cards over. We are required to reverse the order of cards in the deck by such moves.
(a) Prove that for $n = 9$, this can be done in $5$ moves.
(b) Prove that for $n = 52$, this
i. can be done in $27$ moves,
ii. can’t be done in $17$ moves,
iii. can’t be done in $26$ moves.
(SM Voronin, Tchelyabinsk)
2014 India IMO Training Camp, 3
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
2014 Tuymaada Olympiad, 1
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
2016 ASDAN Math Tournament, 5
Let $f(x)$ be a real valued function. Recall that if the inverse function $f^{-1}(x)$ exists, then $f^{-1}(x)$ satisfies $f(f^{-1}(x))=f^{-1}(f(x))=x$. Given that the inverse of the function $f(x)=x^3-12x^2+48x-60$ exists, find all real $a$ that satisfy $f(a)=f^{-1}(a)$.
2019 South East Mathematical Olympiad, 3
$n$ symbols line up in a row, numbered as $1,2,...,n$ from left to right. Delete every symbol with squared numbers. Renumber the rest from left to right. Repeat the process until all $n$ symbols are deleted. Let $f(n)$ be the initial number of the last symbol deleted. Find $f(n)$ in terms of $n$ and find $f(2019)$.
2021 MIG, 5
Kermit writes down the numbers $1$, $2$, $3$, $4$, $5$. He then erases one number, and discovers that the sum of the remaining numbers is $13$. Which number was erased?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
1996 Swedish Mathematical Competition, 3
For every positive integer $n$, we define the function $p_n$ for $x\ge 1$ by
$$p_n(x) = \frac12 \left(\left(x+\sqrt{x^2-1}\right)^n+\left(x-\sqrt{x^2-1}\right)^n\right).$$
Prove that $p_n(x) \ge 1$ and that $p_{mn}(x) = p_m(p_n(x))$.
1983 National High School Mathematics League, 2
Function $f(x)$ is defined on $[0,1]$, $f(0)=f(1)$. For any $x_1,x_2\in [0,1], |f(x_1)-f(x_2)|<|x_1-x_2|(x_1\neq x_2)$. Prove that $|f(x_1)-f(x_2)|<\frac{1}{2}$.
2012 National Olympiad First Round, 15
If $x^4+8x^3+18x^2+8x+a = 0$ has four distinct real roots, then the real set of $a$ is
$ \textbf{(A)}\ (-9,2) \qquad \textbf{(B)}\ (-9,0) \qquad \textbf{(C)}\ [-9,0) \qquad \textbf{(D)}\ [-8,1) \qquad \textbf{(E)}\ (-8,1)$
2019 Polish MO Finals, 3
$n\ge 3$ guests met at a party. Some of them know each other but there is no quartet of different guests $a, b, c, d$ such that in pairs $\lbrace a, b \rbrace, \lbrace b, c \rbrace, \lbrace c, d \rbrace, \lbrace d, a \rbrace$ guests know each other but in pairs $\lbrace a, c \rbrace, \lbrace b, d \rbrace$ guests don't know each other. We say a nonempty set of guests $X$ is an [i]ingroup[/i], when guests from $X$ know each other pairwise and there are no guests not from $X$ knowing all guests from $X$. Prove that there are at most $\frac{n(n-1)}{2}$ different ingroups at that party.
2012 National Olympiad First Round, 13
$20$ points with no three collinear are given. How many obtuse triangles can be formed by these points?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 2{{10}\choose{3}} \qquad \textbf{(D)}\ 3{{10}\choose{3}} \qquad \textbf{(E)}\ {{20}\choose{3}}$
2019 ELMO Shortlist, G5
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
1981 Miklós Schweitzer, 1
We are given an infinite sequence of $ 1$'s and $ 2$'s with the following properties:
(1) The first element of the sequence is $ 1$.
(2) There are no two consecutive $ 2$'s or three consecutive $ 1$'s.
(3) If we replace consecutive $ 1$'s by a single $ 2$, leave the single $ 1$'s alone, and delete the original $ 2$'s, then we recover the original sequence.
How many $ 2$'s are there among the first $ n$ elements of the sequence?
[i]P. P. Palfy[/i]
2009 National Olympiad First Round, 31
For all $ |x| \ge n$, the inequality $ |x^3 \plus{} 3x^2 \minus{} 33x \minus{} 3| \ge 2x^2$ holds. Integer $ n$ can be at least ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 5$
2018 239 Open Mathematical Olympiad, 8-9.7
The sequence $a_n$ is defined by the following conditions: $a_1=1$, and for any $n\in \mathbb N$, the number $a_{n+1}$ is obtained from $a_n$ by adding three if $n$ is a member of this sequence, and two if it is not. Prove that $a_n<(1+\sqrt 2)n$ for all $n$.
[i]Proposed by Mikhail Ivanov[/i]
1998 ITAMO, 1
Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ .
You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.
2022 BMT, 2
Sohom constructs a square $BERK$ of side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$ is parallel to $\overline{BO}$. Compute the area of $CALI$.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/0fda0c273bb73b85f3b1bc73661126630152b3.png[/img]
2018 Auckland Mathematical Olympiad, 2
Starting with a list of three numbers, the “[i]Make-My-Day[/i]” procedure creates a new list by replacing each number by the sum of the other two. For example, from $\{1, 3, 8\}$ “[i]Make-My-Day[/i]” gives $\{11, 9, 4\}$ and a new “[i]MakeMy-Day[/i]” leads to $\{13, 15, 20\}$. If we begin with $\{20, 1, 8\}$, what is the maximum difference between two numbers on the list after $2018$ consecutive “[i]Make-My-Day[/i]”s?
2009 Dutch IMO TST, 3
Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.