Found problems: 85335
2014 District Olympiad, 2
Let $ABC$ be a triangle and let the points $D\in BC, E\in AC, F\in AB$, such that \[ \frac{DB}{DC}=\frac{EC}{EA}=\frac{FA}{FB} \]
The half-lines $AD, BE,$ and $CF$ intersect the circumcircle of $ABC$ at points $M,N$ and $P$. Prove that the triangles $ABC$ and $MNP$ share the same centroid if and only if the areas of the triangles $BMC, CNA$ and $APB$ are equal.
1978 IMO Longlists, 47
Given the expression
\[P_n(x) =\frac{1}{2^n}\left[(x +\sqrt{x^2 - 1})^n+(x-\sqrt{x^2 - 1})^n\right],\]
prove:
$(a) P_n(x)$ satisfies the identity
\[P_n(x) - xP_{n-1}(x) + \frac{1}{4}P_{n-2}(x) \equiv 0.\]
$(b) P_n(x)$ is a polynomial in $x$ of degree $n.$
2021 Novosibirsk Oral Olympiad in Geometry, 4
It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?
2005 Serbia Team Selection Test, 1
Prove that there is n rational number $r$ such that $cosr\pi=\frac{3}{5}$
2014-2015 SDML (Middle School), 2
Suppose $a=1332$ and $b=-222$. Find $c$ such that $\left(\frac{a}{c}\right)^3=\sqrt{b^6}$.
2006 Sharygin Geometry Olympiad, 5
a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip.
b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$).
The strip can be bent, but not torn.
1998 Bundeswettbewerb Mathematik, 3
A triangle $ABC$ satisfies $BC = AC +\frac12 AB$. Point $P$ on side $AB$ is taken so that $AP = 3PB$. Prove that $ \angle PAC = 2\angle CPA$.
2021 Saudi Arabia Training Tests, 27
Each of $N$ people have chosen some $5$ elements from a $23$-element set so that any two people share at most $3$ chosen elements. Does this mean that $N \le 2020$? Answer the same question with $25$ instead of $23$.
2021 Indonesia TST, A
Let $a$ and $b$ be integers. Find all polynomial with integer coefficients sucht that $P(n)$ divides $P(an+b)$ for infinitely many positive integer $n$
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 $a_1$.