Found problems: 85335
Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.
A father, a mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are:
(i) The weakest player chooses the first two contestants.
(ii) The winner of any game plays the next game against the person left out.
(iii) The first person to win two games wins the tournament.
The father is the weakest player, the son the strongest, and it is assumed that any player's probability of winning an individual game from another player does not change during the tournament.
Prove that the father's optimal strategy for winning the tournament is to play the first game with his wife.
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i]
The symbolism $ \lfloor x\rfloor$ denotes the largest integer not exceeding $ x$. For example. $ \lfloor3\rfloor\equal{}3$, and $ \lfloor 9/2\rfloor\equal{}4$. Compute
\[ \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor.
\]$ \textbf{(A)}\ 35 \qquad
\textbf{(B)}\ 38 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 42 \qquad
\textbf{(E)}\ 136$
Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$.
The names of $10$ people are inside $10$ boxes. Each box is labeled with someone's name. Unfortunately, there was a mix-up and not everyone's name is in his or her own box. Each person looks through $5$ boxes in the following way. First, they look in their own box. After looking in a box, they look at the box labeled with the name they found in the previous box. (For example, if someone looks in a box and finds the name "Joe," he or she will then look at the box labeled "Joe.")
What is the probability that every person finds his or her own name in a box?
Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of $ n = 7$ squares.
[img]https://cdn.artofproblemsolving.com/attachments/1/7/c636115180fd624cbeec0c6adda31b4b89ed60.png[/img]
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.
For which $n$ can Bob ensure a win no matter how Alice plays?
For which $n$ can Alice ensure a win no matter how Bob plays?
[i](Karl Czakler)[/i]
A cycling group that has $4n$ members will have several cycling events, such that:
a) Two cycling events are done every week; once on Saturday and once on Sunday.
b) Exactly $2n$ members participate in any cycling event.
c) No member may participate in both cycling events of a week.
d) After all cycling events are completed, the number of events where each pair of members meet is the same for all pairs of members.
Prove that after all cycling events are completed, the number of events where each group of three members meet is the same value $t$ for all groups of three members, and that for $n \ge 2$, $t$ is divisible by $n-1$.
Among 16 coins there are 8 heavy coins with weight of 11 g, and 8 light coins with weight of 10 g, but it's unknown what weight of any coin is. One of the coins is anniversary. How to know, is anniversary coin heavy or light, via three weighings on scales with two cups and without any weight?
On a board the numbers $1,2,3,\dots,98,99$ are written. One has to mark $50$ of them, such that the sum of two marked numbers is never equal to $99$ or $100$. How many ways one can mark these numbers?
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $S(x)$ at the hight $x$.
(2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$
Determine all pairs $(a, b)$ of integers that satisfy both:
1. $5 \leq b < a$
2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order.
[b]Note:[/b] Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.
If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have?
[i](Zhuge Liang)[/i]
Find all positive integers $n$ for which the number
$\frac{n^{3n-2}-3n+1}{3n-2}$
is whole.
[hide=EDIT:] In the original problem instead of whole we search for integers, so with this change $n=1$ will be a solution. [/hide]
The point $M$ belongs to the side $[AC]$ of the acute-angle triangle $ABC$. Two circles are circumscribed around triangles $ABM$ and $BCM$ . What $M$ position corresponds to the minimal area of those circles intersection?
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.
The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point.
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$
Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form
\[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\]
with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.
Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements.
a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1;
b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1.
[i]After an idea of Harazi[/i]
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?
Princess Pear has $100$ jesters with heights $1, 2, \dots, 100$ inches. On day $n$ with $1 \leq n \leq 100$, Princess Pear holds a court with all her jesters with height at most $n$ inches, and she receives two candied cherries from every group of $6$ jesters with a median height of $n - 50$ inches. A jester can be part of multiple groups.
On day $101$, Princess Pear summons all $100$ jesters to court one final time. Every group of $6$ jesters with a median height of 50.5 inches presents one more candied cherry to the Princess. How many candied cherries does Princess Pear receive in total?
Please provide a numerical answer (with justification).
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have \[\frac{|R|}{|P|}\leq \sqrt 2\]
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.
[i]Proposed by Witold Szczechla, Poland[/i]
Determine the greatest natural number $n$, such that for each set $S$ of 2015 different integers there exist 2 subsets of $S$ (possible to be with 1 element and not necessarily non-intersecting) each of which has a sum of its elements divisible by $n$.
There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person.
In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\rightarrow A$, or $A\rightarrow D\rightarrow A\rightarrow D\rightarrow C\rightarrow A\rightarrow B\rightarrow A)$.