Found problems: 85335
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 $x, y \in \Bbb{R}.$
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.