Found problems: 85335
1982 Czech and Slovak Olympiad III A, 5
Given is a sequence of real numbers $\{a_n\}^{\infty}_{n=1}$ such that $a_n \ne a_m$ for $n\ne m,$ given is a natural number $k$. Construct an injective map $P:\{1,2,\ldots,20k\}\to\mathbb Z^+$ such that the following inequalities hold:
$$a_{p(1)}<a_{p(2)}<...<a_{p(10)}$$
$$ a_{p(10)}>a_{p(11)}>...>a_{p(20)}$$
$$a_{p(20)}<a_{p(21)}<...<a_{p(30)}$$
$$...$$
$$a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}$$
$$a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))} $$
$$a_{p(1)}<a_{p(20)}<...<a_{p(20k)},$$
1999 Mongolian Mathematical Olympiad, Problem 5
Let $D$ be a point in the angle $ABC$. A circle $\gamma$ passing through $B$ and $D$ intersects the lines $AB$ and $BC$ at $M$ and $N$ respectively. Find the locus of the midpoint of $MN$ when circle $\gamma$ varies.
1994 AMC 12/AHSME, 6
In the sequence
\[ ..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,... \]
each term is the sum of the two terms to its left. Find $a$.
$ \textbf{(A)}\ -3 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 3 $
2022 Romania National Olympiad, P2
Let $\mathcal{F}$ be the set of pairs of matrices $(A,B)\in\mathcal{M}_2(\mathbb{Z})\times\mathcal{M}_2(\mathbb{Z})$ for which there exists some positive integer $k$ and matrices $C_1,C_2,\ldots, C_k\in\{A,B\}$ such that $C_1C_2\cdots C_k=O_2.$ For each $(A,B)\in\mathcal{F},$ let $k(A,B)$ denote the minimal positive integer $k$ which satisfies the latter property.
[list=a]
[*]Let $(A,B)\in\mathcal{F}$ with $\det(A)=0,\det(B)\neq 0$ and $k(A,B)=p+2$ for some $p\in\mathbb{N}^*.$ Show that $AB^pA=O_2.$
[*]Prove that for any $k\geq 3$ there exists a pair $(A,B)\in\mathcal{F}$ such that $k(A,B)=k.$
[/list][i]Bogdan Blaga[/i]
2023 Moldova Team Selection Test, 5
Find all pairs of positive integers $(n,k)$ for which the number $m=1^{2k+1}+2^{2k+1}+\cdots+n^{2k+1}$ is divisible by $n+2.$
2006 Putnam, B4
Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$.) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$
2013 Stanford Mathematics Tournament, 4
Evaluate $\int_{0}^{4}e^{\sqrt{x}} \, dx$.
2016 Junior Balkan Team Selection Tests - Romania, 2
x,y are real numbers different from 0 such that :$x^3+y^3+3x^2y^2=x^3y^3$
Find all possible values of E=$\dfrac{1}{x}+\dfrac{1}{y}$
2021 JHMT HS, 2
Call a positive integer [i]almost square[/i] if it is not a perfect square, but all of its digits are perfect squares. For example, both $149$ and $904$ are almost square, but $144$ and $936$ are not. Find the number of positive integers less than $1000$ that are not almost square.
2012 AMC 12/AHSME, 22
Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $
2003 Bulgaria Team Selection Test, 3
Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.
2011 Junior Balkan Team Selection Tests - Romania, 3
Let n be a positive integer and let $x_1, x_2,...,x_n$ and $y_1, y_2,...,y_n$ be real numbers. Prove that there exists a number $i, i = 1, 2,...,n$, such that $$\sum_{j=1}^n |x_i - x_j | \le \sum_{j=1}^n |x_i - y_j | $$
2017 Harvard-MIT Mathematics Tournament, 8
Consider all ordered pairs of integers $(a,b)$ such that $1\le a\le b\le 100$ and $$\frac{(a+b)(a+b+1)}{ab}$$ is an integer.
Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose $(3,85)$ over $(2,85)$ over $(4,84)$. Note that your answer should be an ordered pair.
2025 Ukraine National Mathematical Olympiad, 10.2
Given $12$ segments, it is known that they can be divided into $4$ groups of $3$ segments each in such a way that a triangle can be formed from the segments of each triplet. Is it always possible to divide these $12$ segments into $3$ groups of $4$ segments each in such a way that a quadrilateral can be formed from the segments of each quartet?
[i]Proposed by Mykhailo Shtandenko[/i]
1983 IMO Longlists, 47
In a plane, three pairwise intersecting circles $C_1, C_2, C_3$ with centers $M_1,M_2,M_3$ are given. For $i = 1, 2, 3$, let $A_i$ be one of the points of intersection of $C_j$ and $C_k \ (\{i, j, k \} = \{1, 2, 3 \})$. Prove that if $ \angle M_3A_1M_2 = \angle M_1A_2M_3 = \angle M_2A_3M_1 = \frac{\pi}{3}$(directed angles), then $M_1A_1, M_2A_2$, and $M_3A_3$ are concurrent.
2001 Vietnam Team Selection Test, 1
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by
\[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\]
for all $n \in \mathbb{N}^{*}$.
Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.
(Here $[x]$ denotes the integral part of real number $x$).
2009 Mediterranean Mathematics Olympiad, 3
Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled:
i) In every row, the entries add up to the same sum $S$.
ii) In every column, the entries also add up to this sum $S$.
iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2024 Turkey Team Selection Test, 8
For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies
$$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$
Determine the number of divisors of $2024^{2024}$ amongst the sequence.
1997 IMO, 4
An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that:
(a) there is no silver matrix for $ n \equal{} 1997$;
(b) silver matrices exist for infinitely many values of $ n$.
2024 Canadian Mathematical Olympiad Qualification, 3
Let $\vartriangle ABC$ be an acute triangle with $AB < AC$. Let $H$ be its orthocentre and $M$ be the midpoint of arc $BAC$ on the circumcircle. It is given that $B$, $H$, $M$ are collinear, the length of the altitude from $M$ to $AB$ is $1$, and the length of the altitude from $M$ to $BC$ is $6$. Determine all possible areas for $\vartriangle ABC$ .
1952 AMC 12/AHSME, 44
If an integer of two digits is $ k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by:
$ \textbf{(A)}\ (9 \minus{} k) \qquad\textbf{(B)}\ (10 \minus{} k) \qquad\textbf{(C)}\ (11 \minus{} k) \qquad\textbf{(D)}\ (k \minus{} 1) \qquad\textbf{(E)}\ (k \plus{} 1)$
2017 Czech-Polish-Slovak Junior Match, 2
Given is the triangle $ABC$, with $| AB | + | AC | = 3 \cdot | BC | $. Let's denote $D, E$ also points that $BCDA$ and $CBEA$ are parallelograms. On the sides $AC$ and $AB$ sides, $F$ and $G$ are selected respectively so that $| AF | = | AG | = | BC |$. Prove that the lines $DF$ and $EG$ intersect at the line segment $BC$
2003 Swedish Mathematical Competition, 2
In a lecture hall some chairs are placed in rows and columns, forming a rectangle. In each row there are $6$ boys sitting and in each column there are $8$ girls sitting, whereas $15$ places are not taken. What can be said about the number of rows and that of columns?
Kvant 2024, M2823
A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane.
[i]P.A.Kozhevnikov[/i]
2017 Hanoi Open Mathematics Competitions, 14
Put $P = m^{2003}n^{2017} - m^{2017}n^{2003}$ , where $m, n \in N$.
a) Is $P$ divisible by $24$?
b) Do there exist $m, n \in N$ such that $P$ is not divisible by $7$?