Found problems: 85335
1993 All-Russian Olympiad, 4
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].
2011 Sharygin Geometry Olympiad, 18
On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is
a) the minimal,
b) the maximal number of these parts that can be angles?
2016 Federal Competition For Advanced Students, P2, 3
Consider arrangements of the numbers $1$ through $64$ on the squares of an $8\times 8$ chess board, where each square contains exactly one number and each number appears exactly once.
A number in such an arrangement is called super-plus-good, if it is the largest number in its row and at the same time the smallest number in its column. Prove or disprove each of the following statements:
(a) Each such arrangement contains at least one super-plus-good number.
(b) Each such arrangement contains at most one super-plus-good number.
Proposed by Gerhard J. Woeginger
2015 NIMO Summer Contest, 1
For all real numbers $a$ and $b$, let \[a\Join b=\dfrac{a+b}{a-b}.\] Compute $1008\Join 1007$.
[i] Proposed by David Altizio [/i]
1997 Bulgaria National Olympiad, 2
Given a triangle $ABC$.
Let $M$ and $N$ be the points where the angle bisectors of the angles $ABC$ and $BCA$ intersect the sides $CA$ and $AB$, respectively.
Let $D$ be the point where the ray $MN$ intersects the circumcircle of triangle $ABC$.
Prove that $\frac{1}{BD}=\frac{1}{AD}+\frac{1}{CD}$.
2013 Oral Moscow Geometry Olympiad, 4
Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.
2021 CCA Math Bonanza, L5.1
Estimate the number of distinct submissions to this problem. Your submission must be a positive integer less than or equal to $50$. If you submit $E$, and the actual number of distinct submissions is $D$, you will receive a score of $\frac{2}{0.5|E-D|+1}$.
[i]2021 CCA Math Bonanza Lightning Round #5.1[/i]
2023 Brazil Team Selection Test, 2
Find all integers $n$ satisfying $n \geq 2$ and $\dfrac{\sigma(n)}{p(n)-1} = n$, in which $\sigma(n)$ denotes the sum of all positive divisors of $n$, and $p(n)$ denotes the largest prime divisor of $n$.
2018 China Western Mathematical Olympiad, 2
Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$.
Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$
Where $\{x\}$ denotes the fractional part of $x$.
2024 Macedonian TST, Problem 5
Let \(P\) be a convex polyhedron with the following properties:
[b]1)[/b] \(P\) has exactly \(666\) edges.
[b]2)[/b] The degrees of all vertices of \(P\) differ by at most \(1\).
[b]3)[/b] There is an edge‐coloring of \(P\) with \(k\) colors such that for each color \(c\) and any two distinct vertices \(V_1,V_2\), there exists a path from \(V_1\) to \(V_2\) all of whose edges have color \(c\).
Determine the largest positive integer \(k\) for which such a polyhedron \(P\) exists.
JBMO Geometry Collection, 2006
The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.
PEN D Problems, 19
Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]
2003 Tuymaada Olympiad, 3
Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$.
Prove that the set $S$ is finite.
[i]Proposed by F. Bakharev[/i]
1999 Bundeswettbewerb Mathematik, 1
The vertices of a regular $2n$-gon (with $n > 2$ an integer) are labelled with the numbers $1,2,...,2n$ in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if $n$ is odd.
2005 iTest, 40
$ITEST + AHSIMC = 6666CS$. Each letter represents a unique digit from $0$ to $9$. How many solutions of the form $(C,A,S,H)$ exist?
2004 India IMO Training Camp, 2
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
2003 District Olympiad, 3
Consider an array $n \times n$ ($n\ge 2$) with $n^2$ integers. In how many ways one can complete the array if the product of the numbers on any row and column is $5$ or $-5$?
2003 AMC 8, 2
Which of the following numbers has the smallest prime factor?
$\textbf{(A)}\ 55 \qquad
\textbf{(B)}\ 57 \qquad
\textbf{(C)}\ 58 \qquad
\textbf{(D)}\ 59\qquad
\textbf{(E)}\ 61$
2007 Thailand Mathematical Olympiad, 5
A triangle $\vartriangle ABC$ has $\angle A = 90^o$, and a point $D$ is chosen on $AC$. Point $F$ is the foot of altitude from $A$ to $BC$. Suppose that $BD = DC = CF = 2$. Compute $AC$.
1988 IMO Longlists, 18
Let $ N \equal{} \{1,2 \ldots, n\}, n \geq 2.$ A collection $ F \equal{} \{A_1, \ldots, A_t\}$ of subsets $ A_i \subseteq N,$ $ i \equal{} 1, \ldots, t,$ is said to be separating, if for every pair $ \{x,y\} \subseteq N,$ there is a set $ A_i \in F$ so that $ A_i \cap \{x,y\}$ contains just one element. $ F$ is said to be covering, if every element of $ N$ is contained in at least one set $ A_i \in F.$ What is the smallest value $ f(n)$ of $ t,$ so there is a set $ F \equal{} \{A_1, \ldots, A_t\}$ which is simultaneously separating and covering?
2010 Indonesia MO, 4
Given that $m$ and $n$ are positive integers with property:
\[(mn)\mid(m^{2010}+n^{2010}+n)\]
Show that there exists a positive integer $k$ such that $n=k^{2010}$
[i]Nanang Susyanto, Yogyakarta[/i]
2017 Sharygin Geometry Olympiad, P3
Let $I$ be the incenter of triangle $ABC$; $H_B, H_C$ the orthocenters of triangles $ACI$ and $ABI$ respectively; $K$ the touching point of the incircle with the side $BC$. Prove that $H_B, H_C$ and K are collinear.
[i]Proposed by M.Plotnikov[/i]
2006 Bundeswettbewerb Mathematik, 2
Prove that there are no integers $x,y$ for that it is $x^3+y^3=4\cdot(x^2y+xy^2+1)$.
2003 AMC 10, 11
A line with slope $ 3$ intersects a line with slope $ 5$ at the point $ (10, 15)$. What is the distance between the $ x$-intercepts of these two lines?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 20$
2016 Benelux, 3
Find all functions $f :\Bbb{ R}\to \Bbb{Z}$ such that $$\left( f(f(y) - x) \right)^2+ f(x)^2 + f(y)^2 = f(y) \cdot \left( 1 + 2f(f(y)) \right),$$ for all $x, y \in \Bbb{R}.$