Found problems: 85335
1983 USAMO, 5
Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\le q\le n$, contained in the given interval is at most $(n+1)/2$.
2020 European Mathematical Cup, 3
Two types of tiles, depicted on the figure below, are given.
[img]https://wiki-images.artofproblemsolving.com//2/23/Izrezak.PNG[/img]
Find all positive integers $n$ such that an $n\times n$ board consisting of $n^2$ unit squares can be covered without gaps with these two types of tiles (rotations and reflections are allowed) so that no two tiles overlap and no part of any tile covers an area outside the $n\times n$ board. \\
[i]Proposed by Art Waeterschoot[/i]
1998 IberoAmerican Olympiad For University Students, 5
A sequence of polynomials $\{f_n\}_{n=0}^{\infty}$ is defined recursively by $f_0(x)=1$, $f_1(x)=1+x$, and
\[(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, \quad k=1,2,\ldots\]
Prove that $f_k(k)=2^k$ for all $k\geq 0$.
2002 USA Team Selection Test, 1
Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that
\[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]
2012 Today's Calculation Of Integral, 779
Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane.
When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.
PEN H Problems, 22
Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]
1979 Chisinau City MO, 180
It is known that for $0\le x \le 1$ the square trinomial $f (x)$ satisfies the condition $|f(x) | \le 1$. Show that $| f '(0) | \le 8.$
IV Soros Olympiad 1997 - 98 (Russia), 9.10
On the plane there is an image of a circle with a marked center. Let an arbitrary angle be drawn on this plane. Using one ruler, construct the bisector of this angle.
2011 IMAR Test, 2
The area of a convex polygon in the plane is equally shared by the four standard quadrants, and all non-zero lattice points lie outside the polygon. Show that the area of the polygon is less than $4$.
2016 Lusophon Mathematical Olympiad, 3
Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$.
For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?
2007 Romania National Olympiad, 1
Let $a, b, c, d \in \mathbb{N^{*}}$ such that the equation \[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \] has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
2005 AMC 12/AHSME, 22
A sequence of complex numbers $ z_0,z_1,z_2,....$ is defined by the rule
\[ z_{n \plus{} 1} \equal{} \frac {i z_n}{\overline{z_n}}
\]where $ \overline{z_n}$ is the complex conjugate of $ z_n$ and $ i^2 \equal{} \minus{} 1$. Suppose that $ |z_0| \equal{} 1$ and $ z_{2005} \equal{} 1$. How many possible values are there for $ z_0$?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 2005\qquad
\textbf{(E)}\ 2^{2005}$
2007 AMC 10, 22
A finite sequence of three-digit integers has the property that the tens and units digits of each terms are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with terms $ 247$, $ 475$, and $ 756$ and end with the term $ 824$. Let $ S$ be the sum of all the terms in the sequence. What is the largest prime number that always divides $ S$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$
2015 Chile National Olympiad, 4
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$.
2016 China Team Selection Test, 6
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
2001 Saint Petersburg Mathematical Olympiad, 9.6
Find all positive integer solution:
$$k^m+m^n=k^n+1$$
[I]Proposed by V. Frank, F. Petrov[/i]
2016 AMC 12/AHSME, 13
Alice and Bob live $10$ miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is $30^\circ$ from Alice's position and $60^\circ$ from Bob's position. Which of the following is closest to the airplane's altitude, in miles?
$\textbf{(A)}\ 3.5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 4.5 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 5.5$
1984 IMO Longlists, 3
The opposite sides of the reentrant hexagon $AFBDCE$ intersect at the points $K,L,M$ (as shown in the figure). It is given that $AL = AM = a, BM = BK = b$, $CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f$.
[img]http://imgur.com/LUFUh.png[/img]
$(a)$ Given length $a$ and the three angles $\alpha, \beta$ and $\gamma$ at the vertices $A, B,$ and $C,$ respectively, satisfying the condition $\alpha+\beta+\gamma<180^{\circ}$, show that all the angles and sides of the hexagon are thereby uniquely determined.
$(b)$ Prove that
\[\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}\]
Easier version of $(b)$. Prove that
\[(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)\]
1978 Romania Team Selection Test, 2
Suppose that $ k,l $ are natural numbers such that $ \gcd (11m-1,k)=\gcd (11m-1, l) , $ for any natural number $ m. $
Prove that there exists an integer $ n $ such that $ k=11^nl. $
LMT Speed Rounds, 2010.8
How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$
2024 LMT Fall, 17
For positive integers $x$, let$$f(x)=\begin{cases} \frac{f\left(\frac{x}{2}\right)}{2} &\mbox{if }x\mbox{ is even,} \\ 2^{-x} &\mbox{if }x\mbox{ is odd.} \end{cases}$$Find $f(1)+f(2)+f(3)+\dots$.
2010 Turkey Junior National Olympiad, 3
In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.
2024 EGMO, 4
For a sequence $a_1<a_2<\cdots<a_n$ of integers, a pair $(a_i,a_j)$ with $1\leq i<j\leq n$ is called [i]interesting[/i] if there exists a pair $(a_k,a_l)$ of integers with $1\leq k<l\leq n$ such that $$\frac{a_l-a_k}{a_j-a_i}=2.$$ For each $n\geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$.
2007 Croatia Team Selection Test, 4
Given a finite string $S$ of symbols $X$ and $O$, we write $@(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. (For example, $@(XOOXOOX) =-1$.) We call a string $S$ [b]balanced[/b] if every substring $T$ of (consecutive symbols) $S$ has the property $-2 \leq @(T) \leq 2$. (Thus $XOOXOOX$ is not balanced since it contains the sub-string $OOXOO$ whose $@$-value is $-3$.) Find, with proof, the number of balanced strings of length $n$.
2000 Iran MO (3rd Round), 1
Does there exist a natural number $N$ which is a power of$2$, such that one
can permute its decimal digits to obtain a different power of $2$?