Found problems: 85335
The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$.
Find all numbers $n$ such that the value of this expression is $13$.
[list]
[*] There are $128$ coins of two different weights, $64$ each. How can one always find two coins of different weights by performing no more than $7$ weightings on a regular balance?
[*] There are $8$ coins of two different weights, $4$ each. How can one always find two coins of different weights by performing two weightings on a regular balance?[/list]
Prove that if a plane section of a tetrahedron is a parallelogram, then half of its perimeter is contained between the length of the smallest and the length of the largest edge of the tetrahedron.
On the occasion of the 47th Mathematical Olympiad 2016 the numbers 47 and 2016 are written on the blackboard. Alice and Bob play the following game: Alice begins and in turns they choose two numbers $a$ and $b$ with $a > b$ written on the blackboard, whose difference $a-b$ is not yet written on the blackboard and write this difference additionally on the board. The game ends when no further move is possible. The winner is the player who made the last move.
Prove that Bob wins, no matter how they play.
(Richard Henner)
$f(x)$ is a function defined on $\mathbb{R}$. $f(1)=1$, and for all $x\in\mathbb{R}$,
$f(x+5)\geq x+5,f(x+1)\leq f(x)+1$.
If $g(x)=f(x)+1-x$, then $g(2002)=$________.
The point $P$ is outside the circle $\omega$ with center $O$. Lines $\ell_1$ and $\ell_2$ pass through a point $P$, $\ell_1$ touches the circle $\omega$ at the point $A$ and $\ell_2$ intersects $\omega$ at the points $B$ and $C$. Tangent to the circle $\omega$ at points $B$ and $C$ intersect at point $Q$. Let $K$ be the point of intersection of the lines $BC$ and $AQ$. Prove that $(OK) \perp (PQ)$.
On a table there are $100$ red and $k$ white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least once during the moves, the amount of water in these buckets is the same. Find all the possible values of $k$.
$\boxed{\text{G1}}$ Let $M$ be interior point of the triangle $ABC$ with <BAC=70and <ABC=80 If <ACM=10 and <CBM=20.Prove that $AB=MC$
Solve the equation $ \sin^3x \cos 3x \plus{} \cos^3x \sin 3x \equal{} \frac{3}{8}$.
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x.$ Find the value of the sum
\[ \left\lfloor 2+\frac{1}{2^{2021}} \right\rfloor+\left\lfloor 2+\frac{1}{2^{2020}} \right\rfloor+\cdots+\left\lfloor 2+\frac{1}{2^1} \right\rfloor+\left\lfloor 2+\frac{1}{2^0} \right\rfloor. \]
Let $ABC$ be an acute triangle with orthocenter $H$, and let $P$ be a point on the nine-point circle of $ABC$. Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$, respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$, respectively. Show that as $P$ varies along the nine-point circle of $ABC$, the line $QR$ passes through a fixed point.
[i]Proposed by Brandon Wang[/i]
Let $I$ be the incenter in the acute triangle $ABC.$ Rays $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ at points $S$ and $T,$ respectively. The segment $ST$ intersects the sides $AB$ and $AC$ at points $K$ and $L,$ respectively. Prove that $AKIL$ is a rhombus.
In a rhombus, $ ABCD$, line segments are drawn within the rhombus, parallel to diagonal $ BD$, and terminated in the sides of the rhombus. A graph is drawn showing the length of a segment as a function of its distance from vertex $ A$. The graph is:
$ \textbf{(A)}\ \text{A straight line passing through the origin.} \\
\textbf{(B)}\ \text{A straight line cutting across the upper right quadrant.} \\
\textbf{(C)}\ \text{Two line segments forming an upright V.} \\
\textbf{(D)}\ \text{Two line segments forming an inverted V.} \\
\textbf{(E)}\ \text{None of these.}$
Let $x, y, z$ be the real numbers that satisfies the following.
$(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1$
Find the minimum value of $x^4+y^4+z^4$.
Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. At most how many elements does the range set of $f$ have?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 15
\qquad\textbf{(D)}\ 49
\qquad\textbf{(E)}\ \text{Infinitely many}
$
In the triangle $ABC$ the median $AM$ is drawn. Is it possible that the radius of the circle inscribed in $\vartriangle ABM$ could be twice as large as the radius of the circle inscribed in $\vartriangle ACM$ ?
( D . Fomin , Leningrad)
a) Given any positive integer $n$, show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$;
b) If for some positive integers $x$ and $y$, $x+j$ divides $y+j$ for all positive integers $j$, prove that $x = y$.
Consider the function $f_k:\mathbb{Z}^{+}\rightarrow\mathbb{Z}^{+}$ satisfying
\[f_k(x)=x+k\varphi(x)\]
where $\varphi(x)$ is Euler's totient function, that is, the number of positive integers up to $x$ coprime to $x$. We define a sequence $a_1,a_2,...,a_{10}$ with
[list]
[*] $a_1=c$, and
[*] $a_n=f_k(a_{n-1}) \text{ }\forall \text{ } 2\le n\le 10$
[/list]
Is it possible to choose the initial value $c\ne 1$ such that each term is a multiple of the previous, if
(a) $k=2025$ ?
(b) $k=2065$ ?
[i]Proposed by chorn[/i]
A [i]palindrome [/i] is a natural number that works in the decimal system forwards and backwards read is the same size (e.g. $1129211$ or $7337$). Determine all pairs $(m, n)$ of natural numbers, such that
$$(\underbrace{11... 11}_{m}) \cdot (\underbrace{11... 11}_{n})$$ is a palindrome.
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.
Find the smallest possible number of independent deputies after the vacation.
Every cell in a $5\times5$ grid of paper is to be painted either red or white with equal probability. An edge of the paper is said to have a "tree" if the set of cells depicted in the diagram below are all painted red when the paper is rotated so that the edge lies at the bottom. Given that at least one edge of the paper has a tree, what is the expected number of edges that have a tree?
[center][img]https://cdn.artofproblemsolving.com/attachments/1/2/f81d8da53d7bc6819fc1dfe4acb9567d545856.png[/img][/center]
Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$
You flip a fair coin which results in heads ($\text{H}$) or tails ($\text{T}$) with equal probability. What is the probability that you see the consecutive sequence $\text{THH}$ before the sequence $\text{HHH}$?
Find all pairs $(a, b)$ of real numbers with the following property:
[list]Given any real numbers $c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.[/list]
$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$