Found problems: 85335
In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied?
[i]Proposed by Gerhard Wöginger, Austria[/i]
Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.
Let A and B be two rectangles such that it is possible to get rectangle similar to A by putting together rectangles equal to B. Show that it is possible to get rectangle similar to B by putting together rectangles equal to A.
Prove that for all positive integers $n$ the following inequality holds:
$$ \frac{1}{1^2 +2020}+\frac{1}{2^2+2020} + \ldots + \frac{1}{n^2+2020} < \frac{1}{22} $$
Let $a$ and $b$ be two primes having at least two digits, such that $a > b$.
Show that \[240|\left(a^4-b^4\right)\] and show that 240 is the greatest positive integer having this property.
Let $r$ be the inradius of triangle $ABC$. Take a point $D$ on side $BC$, and let $r_1$ and $r_2$ be the inradii of triangles $ABD$ and $ACD$. Prove that $r$, $r_1$, and $r_2$ can always be the side lengths of a triangle.
[b]p1.[/b] James has $8$ Instagram accounts, $3$ Facebook accounts, $4$ QQ accounts, and $3$ YouTube accounts. If each Instagram account has $19$ pictures, each Facebook account has $5$ pictures and $9$ videos, each QQ account has a total of $17$ pictures, and each YouTube account has $13$ videos and no pictures, how many pictures in total does James have in all these accounts?
[b]p2.[/b] If Poonam can trade $7$ shanks for $4$ shinks, and she can trade $10$ shinks for $17$ shenks. How many shenks can Poonam get if she traded all of her $105$ shanks?
[b]p3.[/b] Jerry has a bag with $3$ red marbles, $5$ blue marbles and $2$ white marbles. If Jerry randomly picks two marbles from the bag without replacement, the probability that he gets two different colors can be expressed as a fraction $\frac{m}{n}$ in lowest terms. What is $m + n$?
[b]p4.[/b] Bob's favorite number is between $1200$ and $4000$, divisible by $5$, has the same units and hundreds digits, and the same tens and thousands digits. If his favorite number is even and not divisible by $3$, what is his favorite number?
[b]p5.[/b] Consider a unit cube $ABCDEFGH$. Let $O$ be the center of the face $EFGH$. The length of $BO$ can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ and $b$ are simplified to lowest terms. What is $a + b$?
[b]p6.[/b] Mr. Eddie Wang is a crazy rich boss who owns a giant company in Singapore. Even though Mr. Wang appears friendly, he finds great joy in firing his employees. His immediately fires them when they say "hello" and/or "goodbye" to him. It is well known that $1/2$ of the total people say "hello" and/or "goodbye" to him everyday. If Mr. Wang had $2050$ employees at the end of yesterday, and he hires $2$ new employees at the beginning of each day, in how many days will Mr. Wang first only have $6$ employees left?
[b]p7.[/b] In $\vartriangle ABC$, $AB = 5$, $AC = 6$. Let $D,E,F$ be the midpoints of $\overline{BC}$, $\overline{AC}$, $\overline{AB}$, respectively. Let $X$ be the foot of the altitude from $D$ to $\overline{EF}$. Let $\overline{AX}$ intersect $\overline{BC}$ at $Y$ . Given $DY = 1$, the length of $BC$ is $\frac{p}{q}$ for relatively prime positive integers $p, q$: Find $p + q$.
[b]p8.[/b] Given $\frac{1}{2006} = \frac{1}{a} + \frac{1}{b}$ where $a$ is a $4$ digit positive integer and $b$ is a $6$ digit positive integer, find the smallest possible value of $b$.
[b]p9.[/b] Pocky the postman has unlimited stamps worth $5$, $6$ and $7$ cents. However, his post office has two very odd requirements: On each envelope, an odd number of $7$ cent stamps must be used, and the total number of stamps used must also be odd. What is the largest amount of postage money Pocky cannot make with his stamps, in cents?
[b]p10.[/b] Let $ABCDEF$ be a regular hexagon with side length $2$. Let $G$ be the midpoint of side $DE$. Now let $O$ be the intersection of $BG$ and $CF$. The radius of the circle inscribed in triangle $BOC$ can be expressed in the form $\frac{a\sqrt{b}-\sqrt{c}}{d} $ where $a$, $b$, $c$, $d$ are simplified to lowest terms. What is $a + b + c + d$?
[b]p11.[/b] Estimation (Tiebreaker):
What is the total number of characters in all of the participants' email addresses in the Accuracy Round?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point?
Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.
Find all pairs of integers $ x $, $ y $ satisfying the equation
$$ (xy-1)^2 = (x +1)^2 + (y +1)^2.$$
Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$
$a,b,c \in (0,1)$ and $x,y,z \in ( 0, \infty)$ reals satisfies the condition $a^x=bc,b^y=ca,c^z=ab$. Prove that \[ \dfrac{1}{2+x}+\dfrac{1}{2+y}+\dfrac{1}{2+z} \leq \dfrac{3}{4} \] \\
Let $n \ge 3$ be an odd integer. Each cell is a $n \times n$ board painted in yellow or blue. Let's call the sequence of cells $S_1, S_2,...,S_m$ [i]path [/i] if they are all the same color and the cells $S_i$ and $S_j$ have one in common an edge if and only if $|i - j| = 1$. Suppose that all yellow cells form a path and all the blue cells form a path. Prove that one of the two paths begins or ends at the center of the board.
Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does $p_{10}(x)$ have?
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?
Let there be a triangle $\triangle ABC$ with $BC=a$, $CA=b$, $AB=c$.
Let $T$ be a point not inside $\triangle ABC$ and on the same side of $A$ with respect to $BC$, such that $BT-CT=c-b$.
Let $n=BT$ and $m=CT$. Find the point $P$ that minimizes $f(P)=-a \cdot AP + m \cdot BP + n \cdot CP$.
Consider sequences $a_0$,$a_1$,$a_2$,$\cdots$ of non-negative integers defined by selecting any $a_0$,$a_1$,$a_2$ (not all 0) and for each $n$ $\geq$ 3 letting
$a_n$ = |$a_n-1$ - $a_n-3$|
1-In the particular case that $a_0$ = 1,$a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing
$a_0$,$a_1$,$\cdots$,$a_{19}$,$a_{20}$.
2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$,$a_1$ and $a_2$
3-Prove that, for each choice of $a_0$,$a_1$ and $a_2$, the resulting sequence is eventually periodic.
4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values
$a_0$,$a_1$,$a_2$ of the sequence
In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:
i) $P$ is the circumcentre of triangle $ABC$.
ii) All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.
iii) The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.
Prove that $S$ is the incentre of triangle $ABC$.
We call [i]word [/i] a sequence of letters $\overline {l_1l_2...l_n}, n\ge 1$ .
A [i]word [/i] $\overline {l_1l_2...l_n}, n\ge 1$ is called [i]palindrome [/i] if $l_k=l_{n-k+1}$ , for any $k, 1 \le k \le n$.
Consider a [i]word [/i] $X=\overline {l_1l_2...l_{2014}}$ in which $ l_k\in\{A,B\}$ , for any $k, 1\le k \le 2014$.
Prove that there are at least $806$ [i]palindrome [/i] [i]words [/i] to ''stick" together to get word $X$.
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.
[i]Proposed by Milan Haiman and Carl Schildkraut[/i]
In the acute-angled triangle $ABC$, the segment $AP$ was drawn and the center was marked $O$ of the circumscribed circle. The circumcircle of triangle $ABP$ intersects the line $AC$ for the second time at point $X$, the circumcircle of the triangle $ACP$ intersects the line $AB$ for the second time at the point $Y$. Prove that the lines $XY$ and $PO$ are perpendicular if and only if $P$ is the foor of the bisector of the triangle $ABC$.
A convex $n$-gon is called [i]nice[/i] if its sides are not all the same length, and the sum of the distances of any interior point to the side lines is $1$. Find all integers $n \ge 4$ such that a nice $n$-gon exists .
In $\triangle ABC$, let the bisector of $\angle BAC$ hit the circumcircle at $M$. Let $P$ be the intersection of $CM$ and $AB$. Denote by $(V,WX,YZ)$ the intersection of the line passing $V$ perpendicular to $WX$ with the line $YZ$. Prove that the points $(P,AM,AC), (P,AC,AM), (P,BC,MB)$ are collinear.
[hide=Restatement]In isosceles triangle $APX$ with $AP=AX$, select a point $M$ on the altitude. $PM$ intersects $AX$ at $C$. The circumcircle of $ACM$ intersects $AP$ at $B$. A line passing through $P$ perpendicular to $BC$ intersects $MB$ at $Z$. Show that $XZ$ is perpendicular to $AP$.[/hide]
Carson and Emily attend different schools. Emily’s school has four times as many students as Carson’s school. The total number of students in both schools combined is $10105$. How many students go to Carson’s school?