Found problems: 85335
Two convex quadrilaterals are called [i]partners[/i] if they have three vertices in common and they can be labeled $ABCD$ and $ABCE$ so that $E$ is the reflection of $D$ across the perpendicular bisector of the diagonal $\overline{AC}$. Is there an infinite sequence of convex quadrilaterals such that each quadrilateral is a partner of its successor and no two elements of the sequence are congruent?
[center][img]https://cdn.artofproblemsolving.com/attachments/6/e/cc9da12a49043410c50733cb6843e5ec1005d3.jpeg[/img][/center]
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
In how many rearrangements of the numbers $1, \ 2, \ 3, \ 4, \ 5,\ 6, \ 7, \ 8,\ 9$ do the numbers form a $\textit{hill}$, that is, the numbers form an increasing sequence at the beginning up to a peak, and then form a decreasing sequence to the end such as in $129876543$ or $258976431$?
A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates.
a) Find a latticed rectangle of area $2013$, whose sides are not parallel to the axes.
b) Show that if a latticed rectangle has area $2011$, then their sides are parallel to the axes.
Let $S$ be the set $\{-1, 1\}^n$, that is, $n$-tuples such that each coordinate is either $-1$ or $1$. For \[s = (s_1, s_2, \ldots, s_n), t = (t_1, t_2, \ldots, t_n) \in \{-1, 1\}^n,\] define $s \odot t = (s_1t_1, s_2t_2, \ldots, s_nt_n)$.
Let $c$ be a positive constant, let $f : S \to \{-1, 1\}$ be a function such that there are at least $(1-c) \cdot 2^{2n}$ pairs $(s, t)$ with $s, t \in S$ such that $f(s \odot t) = f(s)f(t)$. Show that there exists a function $f'$ such that $f'(s \odot t) = f'(s)f'(t)$ for all $s, t \in S$ and $f(s) = f'(s)$ for at least $(1-10c) \cdot 2^n$ values of $s \in S$.
How many four-digit positive integers $\overline{a_1a_2a_3a_4}$ have only nonzero digits and have the property that $|a_i-a_j| \neq 1$ for all $1 \leq i<j \leq 4?$
[i]Proposed by Kyle Lee[/i]
Find all positive integers $n$ such that $n!+2$ divides $(2n)!$.
Which bond is strongest?
${ \textbf{(A)}\ \text{C=C}\qquad\textbf{(B)}\ \text{C=N}\qquad\textbf{(C)}\ \text{C=O}\qquad\textbf{(D)}}\ \text{C=S}\qquad $
a) Given a convex hexagon $ABCDEF$, which has a center of symmetry. Prove that the perimeter of triangle $ACE$ is greater than half the perimeter of hexagon $ABCDEF$.
b) Given a convex $(2n)$-gon $P$ having a center of symmetry, its vertices are colored alternately red and blue. Let $Q$ be an $n$-gon with red vertices. Is it possible to say that the perimeter of $Q$ is certainly greater than half the perimeter $P$? Solve the problem for $n = 4$ and $n = 5$.
Let $ K_n(n=1,2,\ldots)$ be periodical continuous functions of period $ 2 \pi$, and write \[ k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .\] Prove that the following statements are equivalent:
(i) $ \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $ f \in \mathcal{L}_1[0,2 \pi]$.
(ii) $ k_n(f;0) \rightarrow f(0)$ for all continuous, $ 2 \pi$-periodic functions $ f$.
[i]V. Totik[/i]
A bag contains plastic cubes of the same size, whose faces have been painted in colors: white, red, yellow, green, blue and violet (without repeating a color on two faces of the same cube). How many of these cubes can there be distinguishable to each other?
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that
\[\min \{|A|,|B|\}\le\log _2n.\]
A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property:
For each $ a\in\mathbb N$, that $ (a,m) \equal{} 1$, has a unique representation in the following form:
\[ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}\]
Prove that for each $ m$ we have a hyper-primitive root.
[b] Problem 6.[/b] Let $m\geq 5$ and $n$ are given natural numbers, and $M$ is regular $2n+1$-gon. Find the number of the convex $m$-gons with vertices among the vertices of $M$, who have at least one acute angle.
[i]Alexandar Ivanov[/i]
Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.
[i]Carl Schildkraut, USA[/i]
Show that
\[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\]
is an integer, for any positive integer \(n\).
Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.
Find all positive integers $x,y,z$ such that $7^x + 13^y = 8^z$
Two different points $A,S$ are given in the plane. Furthermore, positive numbers $d,\omega$ are given, $\omega<180^\circ.$ Let $X$ be a point and $X'$ its image under the rotation by the angle $\omega$ (in counter-clockwise direction) with respect to the origin $S.$ Construct all points $X$ such that $XX'=d$ and $A$ is a point of the segment $XX'.$ Discuss conditions of solvability (in terms of $d,\omega,SA$).
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
[asy]
import three;
size(3inch);
currentprojection=orthographic(-8,15,15);
triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P;
A = (0,0,0);
B = (0,10,0);
C = (12,10,0);
D = (12,0,0);
E = (0,0,5);
F = (0,10,5);
G = (12,10,5);
H = (12,0,5);
I = (1,1,1);
J = (1,9,1);
K = (11,9,1);
L = (11,1,1);
M = (1,1,5);
N = (1,9,5);
O = (11,9,5);
P = (11,1,5);
//outside box far
draw(surface(A--B--C--D--cycle),white,nolight);
draw(A--B--C--D--cycle);
draw(surface(E--A--D--H--cycle),white,nolight);
draw(E--A--D--H--cycle);
draw(surface(D--C--G--H--cycle),white,nolight);
draw(D--C--G--H--cycle);
//inside box far
draw(surface(I--J--K--L--cycle),white,nolight);
draw(I--J--K--L--cycle);
draw(surface(I--L--P--M--cycle),white,nolight);
draw(I--L--P--M--cycle);
draw(surface(L--K--O--P--cycle),white,nolight);
draw(L--K--O--P--cycle);
//inside box near
draw(surface(I--J--N--M--cycle),white,nolight);
draw(I--J--N--M--cycle);
draw(surface(J--K--O--N--cycle),white,nolight);
draw(J--K--O--N--cycle);
//outside box near
draw(surface(A--B--F--E--cycle),white,nolight);
draw(A--B--F--E--cycle);
draw(surface(B--C--G--F--cycle),white,nolight);
draw(B--C--G--F--cycle);
//top
draw(surface(E--H--P--M--cycle),white,nolight);
draw(surface(E--M--N--F--cycle),white,nolight);
draw(surface(F--N--O--G--cycle),white,nolight);
draw(surface(O--G--H--P--cycle),white,nolight);
draw(M--N--O--P--cycle);
draw(E--F--G--H--cycle);
label("10",(A--B),SE);
label("12",(C--B),SW);
label("5",(F--B),W);[/asy]
$\textbf{(A)}\ 204 \qquad \textbf{(B)}\ 280 \qquad \textbf{(C)}\ 320 \qquad \textbf{(D)}\ 340 \qquad \textbf{(E)}\ 600$
Let $ n\ge 2 $ be a natural number and $ A $ be a subset of $ \{1,2,\ldots ,n\} $ having the property that $ x+y $ belongs to $ A $ for any choosing of $ x,y $ such that $ x+y\le n. $
Prove that the arithmetic mean of the elements of $ A $ is at least $ \frac{n+1}{2} . $
Carl only eats food in the shape of equilateral pentagons. Unfortunately, for dinner he receives a piece of steak in the shape of an equilateral triangle. So that he can eat it, he cuts off two corners with straight cuts to form an equilateral pentagon. The set of possible perimeters of the pentagon he obtains is exactly the interval $[a, b)$, where $a$ and $b$ are positive real numbers. Compute $\frac{a}{b}$ .
In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively, and let $F$ be the foot of the altitude through $A$. Show that the line $DE$, the angle bisector of $\angle ACB$ and the circumcircle of $ACF$ pass through a common point.
[b]Alternate version:[/b] In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AB$ and $AC$, respectively. The line $DE$ and the angle bisector of $\angle ACB$ meet at $G$. Show that $\angle AGC$ is a right angle.
Let $ABC$ an acute triangle.
(a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$;
(b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.