Found problems: 85335
2007 China Team Selection Test, 1
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.
II Soros Olympiad 1995 - 96 (Russia), 11.5
Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?
2004 IberoAmerican, 1
It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that:
1: if 2 squares are adjacent then one of them is marked.
2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.
Find the minimun number of squares we most mark.
2008 Postal Coaching, 5
Consider the triangle $ABC$ and the points $D \in (BC),E \in (CA), F \in (AB)$, such that $\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}$. Prove that if the circumcenters of triangles $DEF$ and $ABC$ coincide, then the triangle $ABC$ is equilateral.
2009 All-Russian Olympiad Regional Round, 9.6
Positive integer $m$ is such that the sum of decimal digits of $8^m$ equals 8. Can the last digit of $8^m$ be equal 6?
(Author: V. Senderov)
(compare with http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=431860)
2007 APMO, 5
A regular $ (5 \times 5)$-array of lights is defective, so that toggling the switch for one light causes each adjacent light in the same row and in the same column as well as the light itself to change state, from on to off, or from off to on. Initially all the lights are switched off. After a certain number of toggles, exactly one light is switched on. Find all the possible positions of this light.
Russian TST 2018, P3
For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$
[i]Proposed by Alex Zhai, United States[/i]
2010 Contests, 1
The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex $B$. If the side length of the field is $60$ meters, and each one of the three regions has the same area, how far must the points $M$ and $N$ be from $D$?
Note: the area of each region includes the area the well occupies.
[asy]
pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60);
pathpen=black;
D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle);
D(B--MP("M",M,W));
D(B--MP("N",N,S));
D(CR(B,3));[/asy]
1963 Putnam, A6
Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$, and let $AB$ and $CD$ be any two other chords through $M$. If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$, prove that $M$ is the midpoint of the segment $PQ.$
2011 N.N. Mihăileanu Individual, 4
[b]a)[/b] Prove that there exists an unique sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying
$$ -\text{ctg} x_n=x_n\in\left( (2n+1)\pi /2,(n+1)\pi \right) , $$
for any nonnegative integer $ n. $
[b]b)[/b] Show that $ \lim_{n\to\infty } \left( \frac{x_n}{(n+1)\pi } \right)^{n^2} =e^{-1/\pi^2} . $
[i]Cătălin Zârnă[/i]
1957 Moscow Mathematical Olympiad, 358
The segments of a closed broken line in space are of equal length, and each three consecutive segments are mutually perpendicular. Prove that the number of segments is divisible by $6$.
2004 IMO Shortlist, 4
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]
2012 Federal Competition For Advanced Students, Part 2, 3
Given an equilateral triangle $ABC$ with sidelength 2, we consider all equilateral triangles $PQR$ with sidelength 1 such that
[list]
[*]$P$ lies on the side $AB$,
[*]$Q$ lies on the side $AC$, and
[*]$R$ lies in the inside or on the perimeter of $ABC$.[/list]
Find the locus of the centroids of all such triangles $PQR$.
2021 Lotfi Zadeh Olympiad, 4
Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties:
[list]
[*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$.
[*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$.
[/list]
2014 AMC 12/AHSME, 10
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
$\textbf{(A) }\dfrac{\sqrt3}4\qquad
\textbf{(B) }\dfrac{\sqrt3}3\qquad
\textbf{(C) }\dfrac23\qquad
\textbf{(D) }\dfrac{\sqrt2}2\qquad
\textbf{(E) }\dfrac{\sqrt3}2$
2023 NMTC Junior, P3
Let $a_i (i=1,2,3,4,5,6)$ are reals. The polynomial
$f(x)=a_1+a_2x+a_3x^2+a_4x^3+a_5x^4+a_6a^5+7x^6-4x^7+x^8$ can be factorized into linear factors $x-x_i$ where
$i \in {1,2,3,...,8}$.
Find the possible values of $a_1$.
2013 ELMO Shortlist, 5
Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic.
[i]Proposed by Eric Chen[/i]
1972 IMO Longlists, 41
The ternary expansion $x = 0.10101010\cdots$ is given. Give the binary expansion of $x$.
Alternatively, transform the binary expansion $y = 0.110110110 \cdots$ into a ternary expansion.
1986 Bulgaria National Olympiad, Problem 5
Let $A$ be a fixed point on a circle $k$. Let $B$ be any point on $k$ and $M$ be a point such that $AM:AB=m$ and $\angle BAM=\alpha$, where $m$ and $\alpha$ are given. Find the locus of point $M$ when $B$ describes the circle $k$.
2014 Taiwan TST Round 1, 2
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
2015 South East Mathematical Olympiad, 5
Given two points $E$ and $F$ lie on segment $AB$ and $AD$, respectively. Let the segments $BF$ and $DE$ intersects at point $C$. If it’s known that $AE+EC=AF+FC$, show that $AB+BC=AD+DC$.
PEN K Problems, 1
Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.
2020 Brazil National Olympiad, 1
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.
MathLinks Contest 2nd, 4.1
The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and
$$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$
Prove that $f$ has at least two complex non-real roots.
2010 IFYM, Sozopol, 3
Let $a,b,c$ be integers, $a>0$ and the equation $ax^2-bx+c=0$ has two distinct real roots in the interval $(0,1)$. Find the least possible value of $a$.