Found problems: 85335
2023 ELMO Shortlist, G6
Let \(ABCDEF\) be a convex cyclic hexagon such that quadrilateral \(ABDF\) is a square, and the incenter of \(\triangle ACE\) lines on \(\overline{BF}\). Diagonal \(CE\) intersects diagonals \(BD\) and \(DF\) at points \(P\) and \(Q\), respectively. Prove that the circumcircle of \(\triangle DPQ\) is tangent to \(\overline{BF}\).
[i]Proposed by Elliott Liu[/i]
2014 Hanoi Open Mathematics Competitions, 4
Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square.
(A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above.
2000 Belarus Team Selection Test, 3.2
(a) Prove that $\{n\sqrt3\} >\frac{1}{n\sqrt3}$ for any positive integer $n$.
(b) Is there a constant $c > 1$ such that $\{n\sqrt3\} >\frac{c}{n\sqrt3}$ for all $n \in N$?
2007 Tournament Of Towns, 4
Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop."
[list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot?
[b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to
the empty spot?[/list]
2012 ELMO Shortlist, 7
Consider a graph $G$ with $n$ vertices and at least $n^2/10$ edges. Suppose that each edge is colored in one of $c$ colors such that no two incident edges have the same color. Assume further that no cycles of size $10$ have the same set of colors. Prove that there is a constant $k$ such that $c$ is at least $kn^\frac{8}{5}$ for any $n$.
[i]David Yang.[/i]
2012 Stanford Mathematics Tournament, 1
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?
2012 CHMMC Spring, 4
Let $P(x)$ be a monic polynomial of degree $3$. Suppose that $P(x)$ has remainder $R(x)$ when it is divided by $(x - 1)(x - 4)$ and $2R(x)$ when it is divided by $(x - 2)(x - 3)$. Given that $P(0) = 5$, find $P(5)$.
1988 AMC 8, 25
A [b]palindrome[/b] is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $ \boxed{1:01} $, $ \boxed{12:21} $.
How many times during a 12-hour period will be palindromes?
$ \text{(A)}\ 57\qquad\text{(B)}\ 60\qquad\text{(C)}\ 63\qquad\text{(D)}\ 90\qquad\text{(E)}\ 93 $
2020 LIMIT Category 2, 17
Let $a_n$ denote the angle opposite to the side of length $4n^2$ units in an integer right angled triangle with lengths of sides of the triangle being $4n^2, 4n^4+1$ and $4n^4-1$ where $n \in N$. Then find the value of $\lim_{p \to \infty} \sum_{n=1}^p a_n$
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi $
(D) $\pi/3$
2003 Manhattan Mathematical Olympiad, 1
The polygon ABCDEFG (shown on the right) is a regular octagon. Prove that the area of the rectangle $ADEH$ is one half the area of the whole polygon $ABCDEFGH$.
[asy]
draw((0,1.414)--(1.414,0)--(3.414,0)--(4.828,1.414)--(4.828,3.414)--(3.414,4.828)--(1.414,4.828)--(0,3.414)--(0,1.414));
fill((0,1.414)--(0,3.414)--(4.828,3.414)--(4.828,1.414)--cycle, mediumgrey);
label("$B$",(1.414,0),SW);
label("$C$",(3.414,0),SE);
label("$D$",(4.828,1.414),SE);
label("$E$",(4.828,3.414),NE);
label("$F$",(3.414,4.828),NE);
label("$G$",(1.414,4.828),NW);
label("$H$",(0,3.414),NW);
label("$A$",(0,1.414),SW);
[/asy]
1998 Brazil National Olympiad, 2
Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.
2012 AMC 10, 10
Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
$ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $
2011 Oral Moscow Geometry Olympiad, 3
A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.
1956 AMC 12/AHSME, 48
If $ p$ is a positive integer, then $ \frac {3p \plus{} 25}{2p \minus{} 5}$ can be a positive integer, if and only if $ p$ is:
$ \textbf{(A)}\ \text{at least }3 \qquad\textbf{(B)}\ \text{at least }3\text{ and no more than }35 \qquad\textbf{(C)}\ \text{no more than }35$
$ \textbf{(D)}\ \text{equal to }35 \qquad\textbf{(E)}\ \text{equal to }3\text{ or }35$
2012 European Mathematical Cup, 3
Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$
\[\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}\]
[i]Proposed by Dimitar Trenevski.[/i]
2003 China Team Selection Test, 2
Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.
1989 Tournament Of Towns, (227) 1
Find the number of solutions in positive integers of the equation $\lfloor \frac{x}{2} \rfloor = \lfloor \frac{x}{11} \rfloor +1$
where $\lfloor A\rfloor$ denotes the integer part of the number $A$, e.g. $\lfloor 2.031\rfloor = 2$, $\lfloor 2\rfloor = 2$, etc.
2006 Turkey MO (2nd round), 3
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.
2014 Tournament of Towns., 4
Point L is marked on side BC of triangle ABC so that AL is twice as long as the median CM. Given that angle ALC is equal to 45 degrees prove that AL is perpendicular to CM.
2014 IFYM, Sozopol, 1
A plane is cut into unit squares, each of which is colored in black or white. It is known that each rectangle 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can this plane be colored?
2008 District Olympiad, 2
Let $ S\equal{}\{1,2,\ldots,n\}$ be a set, where $ n\geq 6$ is an integer. Prove that $ S$ is the reunion of 3 pairwise disjoint subsets, with the same number of elements and the same sum of their elements, if and only if $ n$ is a multiple of 3.
VI Soros Olympiad 1999 - 2000 (Russia), 10.3
Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions:
1) $|f(x)|\ge 1$,
2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.
2008 Singapore Team Selection Test, 2
Find all functions $ f : \mathbb R \rightarrow \mathbb R$ such that $ (x \plus{} y)(f(x) \minus{} f(y)) \equal{} (x \minus{}y)f(x \plus{} y)$ for all $ x, y\in \mathbb R$
2010 Bulgaria National Olympiad, 1
Does there exist a number $n=\overline{a_1a_2a_3a_4a_5a_6}$ such that $\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}$ (all bases are $10$) and $n=a^k$ for some positive integers $a,k$ with $k \geq 3 \ ?$
2025 Alborz Mathematical Olympiad, P3
For every positive integer \( n \), do there exist pairwise distinct positive integers \( a_1, a_2, \dots, a_n \) that satisfy the following condition?
For every \( 3 \leq m \leq n \), there exists an \( i \leq m-2 \) such that:
$$
a_m = a_{\gcd(m-1, i)} + \gcd(a_{m-1}, a_i).
$$
Proposed by Alireza Jannati