Found problems: 85335
Consider the set $S = {1,2,3,\ldots , j}$. Let $m(A)$ denote the maximum element of $A$. Prove that
$$\sum_ {A\subseteq S} m(A) = (j-1)2^j +1$$
Given $10$ points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point $A$ among the $10$ points such that:
1. There is an odd number of red segments connected to $A$}
2. The number of red segments connected to each of the other points are all different
Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.
Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.
Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom
In the Gregorian calendar:
(i) years not divisible by $4$ are common years;
(ii) years divisible by $4$ but not by $100$ are leap years;
(iii) years divisible by $100$ but not by $400$ are common years;
(iv) years divisible by $400$ are leap years;
(v) a leap year contains $366$ days; a common year $365$ days.
Prove that the probability that Christmas falls on a Wednesday is not $1/7.$
Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$.
Show that $SO \perp XY$.
Let $n$ be a positive odd integer , $a_1,a_2,\cdots,a_n$ be any permutation of the positive integers $1,2,\cdots,n$ . Prove that :$(a_1-1)(a^2_2-2)(a^3_3-3)\cdots (a^n_n-n)$ is an even number.
Let $ABCD$ be a parallelogram and let $AX$ and $AY$ be the altitudes from $A$ to $CB, CD$, respectively. A line $\ell \perp XY$ bisects $AX$ and meets $AB, BC$ at $K, L$. Similarly, a line $d \perp XY$ bisects $AY$ and meets $DA, DC$ at $P, Q$. Show that the circumcircles of $\triangle BKL$ and $\triangle DPQ$ are tangent to each other.
Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.
Find the minimal $k$ such that every set of $k$ different lines in $\mathbb R^3$ contains either $3$ mutually parallel lines or $3$ mutually intersecting lines or $3$ mutually skew lines.
Let $-2 < x_1 < 2$ be a real number and define $x_2, x_3, \ldots$ by $x_{n+1} = x_n^2-2$ for $n \geq 1$. Assume that no $x_n$ is $0$ and define a number $A$, $0 \leq A \leq 1$ in the following way: The $n^{\text{th}}$ digit after the decimal point in the binary representation of $A$ is a $0$ if $x_1x_2\cdots x_n$ is positive and $1$ otherwise. Prove that $A = \frac{1}{\pi}\cos^{-1}\left(\frac{x_1}{2}\right)$.
[i]Evan O' Dorney.[/i]
Let $ABC$ be an acute triangle. The altitudes $BE$ and $CF$ intersect at the orthocenter $H$, and point $O$ denotes the circumcenter. Point $P$ is chosen so that $\angle APH = \angle OPE = 90^{\circ}$, and point $Q$ is chosen so that $\angle AQH = \angle OQF = 90^{\circ}$. Lines $EP$ and $FQ$ meet at point $T$. Prove that points $A$, $T$, $O$ are collinear.
We say that a function $f: \mathbb{N}\rightarrow\mathbb{N}$ has the $(\mathcal{P})$ property if, for any $y\in\mathbb{N}$, the equation $f(x)=y$ has exactly 3 solutions.
a) Prove that there exist an infinity of functions with the $(\mathcal{P})$ property ;
b) Find all monotonously functions with the $(\mathcal{P})$ property ;
c) Do there exist monotonously functions $f: \mathbb{Q}\rightarrow\mathbb{Q}$ satisfying the $(\mathcal{P})$ property ?
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.
Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of his row. Assume that each box must be placed so that it straddles two lower boxes. Including the one with no boxes, how many arrangements can Mereduth make?
An arbitrary point $M$ is selected in the interior of the segment $AB$. The square $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with segments $AM$ and $MB$ as their respective bases. The circles circumscribed about these squares, with centers $P$ and $Q$, intersect at $M$ and also at another point $N$. Let $N'$ denote the point of intersection of the straight lines $AF$ and $BC$.
a) Prove that $N$ and $N'$ coincide;
b) Prove that the straight lines $MN$ pass through a fixed point $S$ independent of the choice of $M$;
c) Find the locus of the midpoints of the segments $PQ$ as $M$ varies between $A$ and $B$.
Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree $n$. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if $n$ is odd; otherwise Adamu plays first. Prove that:
[list]
[*] Adamu has a winning strategy if $n$ is odd.
[*] Afaafa has a winning strategy if $n$ is even.
[/list]
A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such
that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron [i]antipodal[/i] if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let $A$ be the number of antipodal pairs of vertices, and let $B$ be the number of antipodal pairs of midpoint edges. Determine the difference $A-B$ in terms of the numbers of vertices, edges, and faces.
[i]Proposed by Kei Irei, Japan[/i]
Determine the poles of the inversions that transform four collienar points $A,B, C, D$, aligned in this order, at four points $A' $, $B' $, $C'$ , $D'$ that are vertices of a rectangle, and such that $A'$ and $C'$ are opposite vertices.
In triangle $ABC$, $\angle A=60^{\circ}$. The point $D$ changes on the segment $BC$. Let $O_1,O_2$ be the circumcenters of the triangles $\Delta ABD,\Delta ACD$, respectively. Let $M$ be the meet point of $BO_1,CO_2$ and let $N$ be the circumcenter of $\Delta DO_1O_2$. Prove that, by changing $D$ on $BC$, the line $MN$ passes through a constant point.
A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$
A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time.
At what time(s) does the student have maximum downward velocity?
$\textbf{(A)}$ At all times between $2 s$ and $4 s$
$\textbf{(B)}$ At $4 s$ only
$\textbf{(C)}$ At all times between $4 s$ and $22 s$
$\textbf{(D)}$ At $22 s$ only
$\textbf{(E)}$ At all times between $22 s$ and $24 s$
Chris and Michael play a game on a $5 \times 5$ board, initially containing some black and white counters as shown below:
[img]https://cdn.artofproblemsolving.com/attachments/8/0/42e1a64b3524a0db722c007b8d6b8eddf2d9e5.png[/img]
Chris begins by removing any black counter, and sliding a white counter from an adjacent square onto the empty square. From that point on, the players take turns. Michael slides a black counter onto an adjacent empty square, and Chris does the same with white counters (no more counters are removed). If a player has no legal move, then he loses.
(a) Show that, even if Chris and Michael play cooperatively, the game will come to an end.
(b) Which player has a winning strategy?