Found problems: 85335
1992 Bundeswettbewerb Mathematik, 1
Below the standard representation of a positive integer $n$ is the representation understood by $n$ in the decimal system, where the first digit is different from $0$. Everyone positive integer n is now assigned a number $f(n)$ by using the standard representation of $n$ last digit is placed before the first.
Examples: $f(1992) = 2199$, $f(2000) = 200$.
Determine the smallest positive integer $n$ for which $f(n) = 2n$ holds.
1984 AIME Problems, 4
Let $S$ be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in $S$ is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in $S$?
2012 Belarus Team Selection Test, 2
Let $\Gamma$ be the incircle of an non-isosceles triangle $ABC$, $I$ be it’s center. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$, respectively. Let $A_2 = \Gamma \cap AA_1, M = C_1B_1 \cup AI$, $P$ and $Q$ be the other (different from $A_1, A_2$) intersection points of $A_1M, A_2M$ and $\Gamma$, respectively. Prove that $A, P, Q$ are collinear.
(A. Voidelevich)
1977 AMC 12/AHSME, 25
Determine the largest positive integer $n$ such that $1005!$ is divisible by $10^n$.
$\textbf{(A) }102\qquad\textbf{(B) }112\qquad\textbf{(C) }249\qquad\textbf{(D) }502\qquad \textbf{(E) }\text{none of these}$
2023 Indonesia TST, C
Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower.
Prove that the team that fi nished fourth won exactly two games.
2015 China Girls Math Olympiad, 2
Let $a\in(0,1)$ ,$f(x)=ax^3+(1-4a)x^2+(5a-1)x-5a+3 $ , $g(x)=(1-a)x^3-x^2+(2-a)x-3a-1 $.
Prove that:For any real number $x$ ,at least one of $|f(x)|$ and $|g(x)|$ not less than $a+1$.
2018 All-Russian Olympiad, 4
On the $n\times n$ checker board, several cells were marked in such a way that lower left ($L$) and upper right($R$) cells are not marked and that for any knight-tour from $L$ to $R$, there is at least one marked cell. For which $n>3$, is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?
2024 LMT Fall, B2
A positive $n$ is called [i]sigma rizz[/i] if the sum of its digits is equal to two times the number of digits it has. Find the number of sigma rizz numbers less than $1000.$
2007 Estonia National Olympiad, 2
Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine $ \angle{AQC}$.
1999 Singapore Team Selection Test, 1
Let $M$ and $N$ be two points on the side BC of a triangle $ABC$ such that $BM =MN = NC$. A line parallel to $AC$ meets the segments $AB, AM$ and $AN$ at the points $D, E$ and $F$ respectively. Prove that $EF = 3DE$
2020 Taiwan TST Round 3, 1
Let $\Omega$ be the $A$-excircle of triangle $ABC$, and suppose that $\Omega$ is tangent to lines $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $M$ be the midpoint of segment $EF$. Two more points $P$ and $Q$ are on $\Omega$ such that $EP$ and $FQ$ are both parallel to $DM$. Let $BP$ meet $CQ$ at point $X$. Prove that the line $AM$ is the angle bisector of $\angle XAD$.
[i]Proposed by Shuang-Yen Lee[/i]
2016 India PRMO, 12
Let $S = 1 + \frac{1}{\sqrt2}+ \frac{1}{\sqrt3}+\frac{1}{\sqrt4}+...+ \frac{1}{\sqrt{99}}+ \frac{1}{\sqrt{100}}$ . Find $[S]$.
You may use the fact that $\sqrt{n} < \frac12 (\sqrt{n} +\sqrt{n+1}) <\sqrt{n+1}$ for all integers $n \ge 1$.
2023 LMT Spring, 3
Phoenix is counting positive integers starting from $1$. When he counts a perfect square greater than $1$, he restarts at $1$, skipping that square the next time. For example, the first $10$ numbers Phoenix counts are $1$, $2$, $3$, $4$, $1$, $2$, $3$, $5$, $6$, $7$, $...$ How many numbers will Phoenix have counted after counting 1$00$ for the first time?
2008 India National Olympiad, 3
Let $ A$ be a set of real numbers such that $ A$ has at least four elements. Suppose $ A$ has the property that $ a^2 \plus{} bc$ is a rational number for all distinct numbers $ a,b,c$ in $ A$. Prove that there exists a positive integer $ M$ such that $ a\sqrt{M}$ is a rational number for every $ a$ in $ A$.
2024 Auckland Mathematical Olympiad, 8
There are $25$ points on the plane, and among any three of them there are two at a distance less than $1$. Prove that there is a circle of radius $1$ containing at least $13$ of these points.
2020 Harvard-MIT Mathematics Tournament, 2
Find the unique pair of positive integers $(a,b)$ with $a< b$ for which
\[\frac{2020-a}{a}\cdot \frac{2020-b}{b}=2.\]
[i]Proposed by James Lin.[/i]
2009 Sharygin Geometry Olympiad, 4
Given regular $17$-gon $A_1 ... A_{17}$. Prove that two triangles formed by lines $A_1A_4, A_2A_{10}, A_{13}A_{14}$ and $A_2A_3, A_4A_6 A_{14}A_{15} $ are equal.
(N.Beluhov)
2017 IMO Shortlist, G3
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
1998 Miklós Schweitzer, 4
For any measurable set $H \subset R$ , we define the sequence $a_n(H)$ by the formula:
$$a_n(H) = \lambda \bigg([0,1] \setminus \bigcup_{k = n}^{2n} (H + \log_2 k) \bigg)$$
where $\lambda$ denotes the Lebesgue measure and $\log_2$ denotes the binary logarithm. Prove that there is a measurable, 1-periodic, positive measure set $H \subset R$ , such that the sequence $a_n( H )$ does not belong to any space $l_p$ ($1 \leq p < \infty$).
[hide=not sure about this part]For what numbers $1 \leq p <\infty$ is it true that whenever H is 1-periodic, positive measure, the sequence $a_n( H )$ belongs to the space $l_p$?[/hide]
2015 Saudi Arabia IMO TST, 1
Find all functions $f : R_{>0} \to R$ such that $f \left(\frac{x}{y}\right) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in R_{>0}$. Here, $R_{>0}$ denotes the set of all positive real numbers.
Nguyễn Duy Thái Sơn
2022/2023 Tournament of Towns, P1
There are two letter sequences $A$ and $B$, both with length $100$ letters. In one move you can insert in any place of sequence ( possibly to start or to end) any number of same letters or remove any number of consecutive same letters.
Prove that it is possible to make second sequence from first sequence using not more than $100$ moves.
2008 Brazil Team Selection Test, 1
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i]
DMM Individual Rounds, 2022 Tie
[b]p1.[/b] The sequence $\{x_n\}$ is defined by $$x_{n+1} = \begin{cases} 2x_n - 1, \,\, if \,\, \frac12 \le x_n < 1 \\ 2x_n, \,\, if \,\, 0 \le x_n < \frac12 \end{cases}$$ where $0 \le x_0 < 1$ and $x_7 = x_0$. Find the number of sequences satisfying these conditions.
[b]p2.[/b] Let $M = \{1, . . . , 2022\}$. For any nonempty set $X \subseteq M$, let $a_X$ be the sum of the maximum and the minimum number of $X$. Find the average value of $a_X$ across all nonempty subsets $X$ of $M$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2008 Putnam, B5
Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)
2009 APMO, 2
Let $ a_1$, $ a_2$, $ a_3$, $ a_4$, $ a_5$ be real numbers satisfying the following equations:
$ \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2}$ for $ k \equal{} 1, 2, 3, 4, 5$
Find the value of $ \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41}$ (Express the value in a single fraction.)