Found problems: 85335
Durer Math Competition CD Finals - geometry, 2015.D1
From all three vertices of triangle $ABC$, we set perpendiculars to the exterior and interior of the other vertices angle bisectors. Prove that the sum of the squares of the segments thus obtained is exactly $2 (a^2 + b^2 + c^2)$, where $a, b$, and $c$ denote the lengths of the sides of the triangle.
2004 Putnam, B4
Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.
2015 IMO Shortlist, C5
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:
(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$.
Prove that there exist two positive integers $b$ and $N$ for which\[\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2\]for all integers $m$ and $n$ such that $n>m\ge N$.
[i]Proposed by Ivan Guo and Ross Atkins, Australia[/i]
2013 International Zhautykov Olympiad, 3
Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that
\[\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.\]
[i]Proposed by Orif Ibrogimov, Uzbekistan.[/i]
2014 Sharygin Geometry Olympiad, 2
A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.
(A. Zertsalov, D. Skrobot)
2022 Thailand TSTST, 2
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2018 Singapore Senior Math Olympiad, 2
In a convex quadrilateral $ABCD, \angle A < 90^o, \angle B < 90^o$ and $AB > CD$. Points $P$ and $Q$ are on the segments $BC$ and $AD$ respectively. Suppose the triangles $APD$ and $BQC$ are similar. Prove that $AB$ is parallel to $CD$.
2022 AIME Problems, 2
Find the three-digit positive integer $\underline{a} \ \underline{b} \ \underline{c}$ whose representation in base nine is $\underline{b} \ \underline{c} \ \underline{a}_{\hspace{.02in}\text{nine}}$, where $a$, $b$, and $c$ are (not necessarily distinct) digits.
1991 AMC 12/AHSME, 26
An $n$-digit positive integer is [i]cute[/i] if its $n$ digits are an arrangement of the set $\{1,2,\ldots,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,\ldots,n$. For example 321 is a cute 3-digit integer because 1 divides 3, 2 divides 32, and 3 divides 321. How many cute 6-digit integers are there?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4 $
2010 Today's Calculation Of Integral, 663
Given are the curve $y=x^2+x-2$ and a curve which is obtained by tranfering the curve symmetric with respect to the point $(p,\ 2p)$. Let $p$ change in such a way that these two curves intersects, find the maximum area of the part bounded by these curves.
[i]1978 Nagasaki University entrance exam/Economics[/i]
2023-24 IOQM India, 22
In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective opposite vertex by a line segment. If $N$ denotes the number of ways we can choose the pegs such that the drawn line segments divide the interior of the triangle into exactly nine regions, find the sum of the squares of the digits of $N$.
PEN S Problems, 27
Which integers have the following property? If the final digit is deleted, the integer is divisible by the new number.
2016 Czech-Polish-Slovak Junior Match, 6
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia
2009 Purple Comet Problems, 11
Aisha went shopping. At the first store she spent $40$ percent of her money plus four dollars. At the second store she spent $50$ percent of her remaining money plus $5$ dollars. At the third store she spent $60$ percent of her remaining money plus six dollars. When Aisha was done shopping at the three stores, she had two dollars left. How many dollars did she have with her when she started shopping?
2022 VN Math Olympiad For High School Students, Problem 5
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that:
a) ${5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).$
b) ${F_{p - 1}} \equiv 0(\bmod p).$
c) $k(p)|p-1.$
2022 Grosman Mathematical Olympiad, P7
Let $k\leq n$ be two positive integers. $n$ points are marked on a line. It is known that for each marked point, the number of marked points at a distance $\leq 1$ from it (including the point itself) is divisible by $k$.
Show that $k$ divides $n$ (without remainder).
2018 Israel National Olympiad, 3
Determine the minimal and maximal values the expression $\frac{|a+b|+|b+c|+|c+a|}{|a|+|b|+|c|}$ can take, where $a,b,c$ are real numbers.
1997 Cono Sur Olympiad, 4
Consider a board with $n$ rows and $4$ columns. In the first line are written $4$ zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed:
* if in the previous line there was a $0$, then in the down square $1$ is placed;
* if in the previous line there was a $1$, then in the down square $2$ is placed;
* if in the previous line there was a $2$, then in the down square $0$ is placed;
Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.
II Soros Olympiad 1995 - 96 (Russia), 11.4
Draw on the coordinate plane a set of points $M(a, b)$ such that the equation $x^4+ax+b=0$ has a unique root satisfying the condition $0 \le x \le 1$.
2021 Grand Duchy of Lithuania, 2
Every number in the sequence $1, 2, ... , 2021$ is either white or black. At one step Alice can choose three numbers of the sequence and change the color of each of them (white to black and black to white) if one of those three numbers is the arithmetic mean of the other two. Alice wants to perform several steps so that at the end all the numbers in the
sequence are black. For which initial colorings of numbers can Alice achieve this?
2001 Brazil Team Selection Test, Problem 4
Prove that for all integers $n\ge3$ there exists a set $A_n=\{a_1,a_2,\ldots,a_n\}$ of $n$ distinct natural numbers such that, for each $i=1,2,\ldots,n$,
$$\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.$$
2020 Harvest Math Invitational Team Round Problems, HMI Team #6
6. A triple of integers $(a,b,c)$ is said to be $\gamma$[i]-special[/i] if $a\le \gamma(b+c)$, $b\le \gamma(c+a)$ and $c\le\gamma(a+b)$. For each integer triple $(a,b,c)$ such that $1\le a,b,c \le 20$, Kodvick writes down the smallest value of $\gamma$ such that $(a,b,c)$ is $\gamma$-special. How many distinct values does he write down?
[i]Proposed by winnertakeover[/i]
2021 Irish Math Olympiad, 4
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the remainder when $m$ is divided by $19$?
2007 Serbia National Math Olympiad, 2
Triangle $\Delta GRB$ is dissected into $25$ small triangles as shown. All vertices of these triangles are painted in three colors so that the following conditions are satisfied: Vertex $G$ is painted in green, vertex $R$ in red, and $B$ in blue; Each vertex on side $GR$ is either green or red, each vertex on $RB$ is either red or blue, and each vertex on $GB$ is either green or blue. The vertices inside the big triangle are arbitrarily colored.
Prove that, regardless of the way of coloring, at least one of the $25$ small triangles has vertices of three different colors.
2022 Germany Team Selection Test, 3
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]