Found problems: 85335
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 $x, y \in \Bbb{R}.$
Russian TST 2015, P2
Let $p\geqslant 5$ be a prime number. Prove that the set $\{1,2,\ldots,p - 1\}$ can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo $p{}$.
2014 Saudi Arabia IMO TST, 4
Points $A_1,~ B_1,~ C_1$ lie on the sides $BC,~ AC$ and $AB$ of a triangle $ABC$, respectively, such that $AB_1 -AC_1 = CA_1 -CB_1 = BC_1 -BA_1$. Let $I_A,~ I_B,~ I_C$ be the incenters of triangles $AB_1C_1,~ A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcenter of triangle $I_AI_BI_C$, is the incenter of triangle $ABC$.
1989 Mexico National Olympiad, 6
Determine the number of paths from $A$ to $B$ on the picture that go along gridlines only, do not pass through any point twice, and never go upwards?
[img]https://cdn.artofproblemsolving.com/attachments/0/2/87868e24a48a2e130fb5039daeb85af42f4b9d.png[/img]
2008 ISI B.Stat Entrance Exam, 10
Two subsets $A$ and $B$ of the $(x,y)$-plane are said to be [i]equivalent[/i] if there exists a function $f: A\to B$ which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.
1977 IMO Longlists, 26
Let $p$ be a prime number greater than $5.$ Let $V$ be the collection of all positive integers $n$ that can be written in the form $n = kp + 1$ or $n = kp - 1 \ (k = 1, 2, \ldots).$ A number $n \in V$ is called [i]indecomposable[/i] in $V$ if it is impossible to find $k, l \in V$ such that $n = kl.$ Prove that there exists a number $N \in V$ that can be factorized into indecomposable factors in $V$ in more than one way.
2011 Tournament of Towns, 1
$P$ and $Q$ are points on the longest side $AB$ of triangle $ABC$ such that $AQ = AC$ and $BP = BC$. Prove that the circumcentre of triangle $CPQ$ coincides with the incentre of triangle $ABC$.
IV Soros Olympiad 1997 - 98 (Russia), 10.2
Solve the equation $$\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}$$
2019 CCA Math Bonanza, TB3
For $i=1,2,\ldots,7$, Zadam Heng chooses a positive integer $m_i$ at random such that each positive integer $k$ is chosen with probability $\frac{2^i-1}{2^{ik}}$. If $m_1+2m_2+\ldots+7m_7\neq35$, Zadam keeps rechoosing the $m_i$ until this equality holds. Given that he eventually stops, what is the probability that $m_4=1$ when Zadam stops?
[i]2019 CCA Math Bonanza Tiebreaker Round #3[/i]
2013 USA TSTST, 7
A country has $n$ cities, labelled $1,2,3,\dots,n$. It wants to build exactly $n-1$ roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly $1$, and it is also not permitted to put a road between cities $1$ and $n$. Let $T_n$ be the total number of possible ways to build these roads.
(a) For all odd $n$, prove that $T_n$ is divisible by $n$.
(b) For all even $n$, prove that $T_n$ is divisible by $n/2$.
Kvant 2019, M2580
We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic.
[i] Tibor Bakos and Géza Kós [/i]
2019 Kazakhstan National Olympiad, 6
The tangent line $l$ to the circumcircle of an acute triangle $ABC$ intersects the lines $AB, BC$, and $CA$ at points $C', A'$ and $B'$, respectively. Let $H$ be the orthocenter of a triangle $ABC$. On the straight lines A'H, B′H and C'H, respectively, points $A_1, B_1$ and $C_1$ (other than $H$) are marked such that $AH = AA_1, BH = BB_1$ and $CH = CC_1$. Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.
2003 Junior Balkan Team Selection Tests - Romania, 2
Two circles $C_1(O_1)$ and $C_2(O_2)$ with distinct radii meet at points $A$ and $B$. The tangent from $A$ to $C_1$ intersects the tangent from $B$ to $C_2$ at point $M$. Show that both circles are seen from $M$ under the same angle.