Found problems: 85335
2015 Math Prize for Girls Problems, 18
Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$.
2000 Harvard-MIT Mathematics Tournament, 12
Calculate the number of ways of choosing $4$ numbers from the set ${1,2,\cdots ,11}$ such that at least $2$ of the numbers are consecutive.
2022 Adygea Teachers' Geometry Olympiad, 3
The incircle of triangle $ABC$ touches its sides at points $A'$, $B'$, $C'$. $I$ is its center. Straight line $B'I$ intersects segment $A'C'$ at point $P$. Prove that straight line $BP$ passes through the midpoint of $AC$.
2021 Kyiv City MO Round 1, 11.4
For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression:
$$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$
[i]Proposed by Serhii Torba[/i]
1990 Romania Team Selection Test, 7
The sequence $ (x_n)_{n \geq 1}$ is defined by:
$ x_1\equal{}1$
$ x_{n\plus{}1}\equal{}\frac{x_n}{n}\plus{}\frac{n}{x_n}$
Prove that $ (x_n)$ increases and $ [x_n^2]\equal{}n$.
2013 NIMO Problems, 14
Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962.
\] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$.
[i]Proposed by Evan Chen[/i]
2006 APMO, 3
Let $p\ge5$ be a prime and let $r$ be the number of ways of placing $p$ checkers on a $p\times p$ checkerboard so that not all checkers are in the same row (but they may all be in the same column). Show that $r$ is divisible by $p^5$. Here, we assume that all the checkers are identical.
2019 Thailand TST, 2
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
[/list]
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
2001 AMC 8, 1
Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to paint one dimple, how many minutes will he need to do his job?
$ \text{(A)}\ 4\qquad\text{(B)}\ 6\qquad\text{(C)}\ 8\qquad\text{(D)}\ 10\qquad\text{(E)}\ 12 $
2018 Turkey Team Selection Test, 8
For integers $m\geq 3$, $n$ and $x_1,x_2, \ldots , x_m$ if $x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n) $ for every $2\leq i \leq m-1$, we say that the $m$-tuple $(x_1,x_2,\ldots , x_m)$ is an arithmetic sequence in $(mod n)$. Let $p\geq 5$ be a prime number and $1<a<p-1$ be an integer. Let ${a_1,a_2,\ldots , a_k}$ be the set of all possible remainders when positive powers of $a$ are divided by $p$. Show that if a permutation of ${a_1,a_2,\ldots , a_k}$ is an arithmetic sequence in $(mod p)$, then $k=p-1$.
1999 AMC 12/AHSME, 24
Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords are the sides of a convex quadrilateral?
$ \textbf{(A)}\ \frac{1}{15}\qquad
\textbf{(B)}\ \frac{1}{91}\qquad
\textbf{(C)}\ \frac{1}{273}\qquad
\textbf{(D)}\ \frac{1}{455}\qquad
\textbf{(E)}\ \frac{1}{1365}$
2016 JBMO Shortlist, 1
Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.
2021 Girls in Mathematics Tournament, 3
A natural number is called [i]chaotigal [/i] if it and its successor both have the sum of their digits divisible by $2021$. How many digits are in the smallest chaotigal number?
2009 Math Prize For Girls Problems, 18
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$, where $ a$, $ b$, and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$?
2001 Romania Team Selection Test, 1
Find all polynomials with real coefficients $P$ such that
\[ P(x)P(2x^2-1)=P(x^2)P(2x-1)\]
for every $x\in\mathbb{R}$.
2021 MOAA, 10
For how many nonempty subsets $S \subseteq \{1, 2, \ldots , 10\}$ is the sum of all elements in $S$ even?
[i]Proposed by Andrew Wen[/i]
2003 Estonia Team Selection Test, 2
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$.
(H. Nestra)
2006 AMC 12/AHSME, 10
For how many real values of $ x$ is $ \sqrt {120 \minus{} \sqrt {x}}$ an integer?
$ \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11$
1983 Canada National Olympiad, 1
Find all positive integers $w$, $x$, $y$ and $z$ which satisfy $w! = x! + y! + z!$.
2015 China Girls Math Olympiad, 7
Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$
1989 National High School Mathematics League, 13
$a_1,a_2,\cdots,a_n$ are positive numbers, satisfying that $a_1a_2\cdots a_n=1$.
Prove that $(2+a_1)(2+a_2)\cdots(2+a_n)\geq3^n$
1989 AMC 8, 22
The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list:
\[\begin{tabular}[t]{lccc}
& & AJHSME & 1989 \\
& & & \\
1. & & JHSMEA & 9891 \\
2. & & HSMEAJ & 8919 \\
3. & & SMEAJH & 9198 \\
& & ........ &
\end{tabular}\]
What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time?
$\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$
Cono Sur Shortlist - geometry, 2020.G1.4
Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.
1990 IMO Longlists, 92
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:
[b][i]i)[/i][/b] Any two lines are non-parallel.
[b][i]ii)[/i][/b] Any three lines are non-concurrent.
[b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$
Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
2012 Mathcenter Contest + Longlist, 6
Let $a,b,c>0$ and $abc=1$. Prove that $$\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.$$
[i](Zhuge Liang)[/i]