Found problems: 85335
2020 Iran MO (3rd Round), 2
Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)
2009 Romania National Olympiad, 2
Let be a real number $ a\in \left[ 2+\sqrt 2,4 \right] . $ Find $ \inf_{\stackrel{z\in\mathbb{C}}{|z|\le 1}} \left| z^2-az+a \right| . $
2017 Saudi Arabia IMO TST, 1
Let $a, b$ and $c$ be positive real numbers such that min $\{ab, bc, ca\} \ge 1$. Prove that
$$\sqrt[3]{(a^2 + 1)(b^2 + 1)(c^2 + 1)} \le (\frac{a+b+c}{3} )^2 + 1 $$
2019 Switzerland Team Selection Test, 4
Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:
$(a)$ $P(x)>x$ for all positive integers $x$.
$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.
2014 Federal Competition For Advanced Students, 4
We are given a right-angled triangle $MNP$ with right angle in $P$. Let $k_M$ be the circle with center $M$ and radius $MP$, and let $k_N$ be the circle with center $N$ and radius $NP$. Let $A$ and $B$ be the common points of $k_M$ and the line $MN$, and let $C$ and $D$ be the common points of $k_N$ and the line $MN$ with with $C$ between $A$ and $B$. Prove that the line $PC$ bisects the angle $\angle APB$.
1995 All-Russian Olympiad Regional Round, 10.5
Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?
2005 QEDMO 1st, 6 (U1)
Prove that for any four real numbers $a$, $b$, $c$, $d$, the inequality
\[ \left(a-b\right)\left(b-c\right)\left(c-d\right)\left(d-a\right)+\left(a-c\right)^2\left(b-d\right)^2\geq 0 \]
holds.
[hide="comment"]This is inequality (350) in: Mihai Onucu Drimbe, [i]Inegalitati, idei si metode[/i], Zalau: Gil, 2003.
Posted here only for the sake of completeness; in fact, it is more or less the same as http://www.mathlinks.ro/Forum/viewtopic.php?t=3152 .[/hide]
Darij
1973 AMC 12/AHSME, 19
Define $ n_a!$ for $ n$ and $ a$ positive to be
\[ n_a ! \equal{} n (n\minus{}a)(n\minus{}2a)(n\minus{}3a)...(n\minus{}ka)\]
where $ k$ is the greatest integer for which $ n>ka$. Then the quotient $ 72_8!/18_2!$ is equal to
$ \textbf{(A)}\ 4^5 \qquad
\textbf{(B)}\ 4^6 \qquad
\textbf{(C)}\ 4^8 \qquad
\textbf{(D)}\ 4^9 \qquad
\textbf{(E)}\ 4^{12}$
2004 Mexico National Olympiad, 6
What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$, if the path does not cross the same place twice?.
2021 Turkey MO (2nd round), 1
Initially, one of the two boxes on the table is empty and the other contains $29$ different colored marbles. By starting with the full box and performing moves in order, in each move, one or more marbles are selected from that box and transferred to the other box. At most, how many moves can be made without selecting the same set of marbles more than once?
2014-2015 SDML (High School), 8
What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
$\text{(A) }ab\frac{3\sqrt{3}}{4}\qquad\text{(B) }ab\qquad\text{(C) }ab\sqrt{2}\qquad\text{(D) }\left(a+b\right)\frac{3\sqrt{3}}{4}\qquad\text{(E) }\left(a+b\right)\sqrt{2}$
2013 Rioplatense Mathematical Olympiad, Level 3, 4
Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game.
For each positive integer $n$, find a winning strategy for one of the players.
2014 AMC 10, 3
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?
$ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $
2003 Rioplatense Mathematical Olympiad, Level 3, 3
An $8\times 8$ chessboard is to be tiled (i.e., completely covered without overlapping) with pieces of the following shapes:
[asy]
unitsize(.6cm);
draw(unitsquare,linewidth(1));
draw(shift(1,0)*unitsquare,linewidth(1));
draw(shift(2,0)*unitsquare,linewidth(1));
label("\footnotesize $1\times 3$ rectangle",(1.5,0),S);
draw(shift(8,1)*unitsquare,linewidth(1));
draw(shift(9,1)*unitsquare,linewidth(1));
draw(shift(10,1)*unitsquare,linewidth(1));
draw(shift(9,0)*unitsquare,linewidth(1));
label("\footnotesize T-shaped tetromino",(9.5,0),S);
[/asy] The $1\times 3$ rectangle covers exactly three squares of the chessboard, and the T-shaped tetromino covers exactly four squares of the chessboard. [list](a) What is the maximum number of pieces that can be used?
(b) How many ways are there to tile the chessboard using this maximum number of pieces?[/list]
2002 Cono Sur Olympiad, 2
Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.
2002 All-Russian Olympiad, 1
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
1982 IMO, 1
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$.
PEN O Problems, 50
What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another?
2014 Korea National Olympiad, 1
For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.
2018 ABMC, 2018 Nov
[b]p1.[/b] How many lines of symmetry does a square have?
[b]p2.[/b] Compute$ 1/2 + 1/6 + 1/12 + 1/4$.
[b]p3.[/b] What is the maximum possible area of a rectangle with integer side lengths and perimeter $8$?
[b]p4.[/b] Given that $1$ printer weighs $400000$ pennies, and $80$ pennies weighs $2$ books, what is the weight of a printer expressed in books?
[b]p5.[/b] Given that two sides of a triangle are $28$ and $3$ and all three sides are integers, what is the sum of the possible lengths of the remaining side?
[b]p6.[/b] What is half the sum of all positive integers between $1$ and $15$, inclusive, that have an even number of positive divisors?
[b]p7.[/b] Austin the Snowman has a very big brain. His head has radius $3$, and the volume of his torso is one third of his head, and the volume of his legs combined is one third of his torso. If Austin's total volume is $a\pi$ where $a$ is an integer, what is $a$?
[b]p8.[/b] Neethine the Kiwi says that she is the eye of the tiger, a fighter, and that everyone is gonna hear her roar. She is standing at point $(3, 3)$. Neeton the Cat is standing at $(11,18)$, the farthest he can stand from Neethine such that he can still hear her roar. Let the total area of the region that Neeton can stand in where he can hear Neethine's roar be $a\pi$ where $a$ is an integer. What is $a$?
[b]p9.[/b] Consider $2018$ identical kiwis. These are to be divided between $5$ people, such that the first person gets $a_1$ kiwis, the second gets $a_2$ kiwis, and so forth, with $a_1 \le a_2 \le a_3 \le a_4 \le a_5$. How many tuples $(a_1, a_2, a_3, a_4, a_5)$ can be chosen such that they form an arithmetic sequence?
[b]p10.[/b] On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$?
[b]p11.[/b] In equiangular hexagon $ABCDEF$, $AB = 7$, $BC = 3$, $CD = 8$, and $DE = 5$. The area of the hexagon is in the form $\frac{a\sqrt{b}}{c}$ with $b$ square free and $a$ and $c$ relatively prime. Find $a+b+c$ where $a, b,$ and $c$ are integers.
[b]p12.[/b] Let $\frac{p}{q} = \frac15 + \frac{2}{5^2} + \frac{3}{5^3} + ...$ . Find $p + q$, where $p$ and $q$ are relatively prime positive integers.
[b]p13.[/b] Two circles $F$ and $G$ with radius $10$ and $4$ respectively are externally tangent. A square $ABMC$ is inscribed in circle $F$ and equilateral triangle $MOP$ is inscribed in circle $G$ (they share vertex $M$). If the area of pentagon $ABOPC$ is equal to $a + b\sqrt{c}$, where $a$, $b$, $c$ are integers $c$ is square free, then find $a + b + c$.
[b]p14.[/b] Consider the polynomial $P(x) = x^3 + 3x^2 + ax + 8$. Find the sum of all integer $a$ such that the sum of the squares of the roots of $P(x)$ divides the sum of the coecients of $P(x)$.
[b]p15.[/b] Nithin and Antonio play a number game. At the beginning of the game, Nithin picks a prime $p$ that is less than $100$. Antonio then tries to find an integer $n$ such that $n^6 + 2n^5 + 2n^4 + n^3 + (n^2 + n + 1)^2$ is a multiple of $p$. If Antonio can find such a number n, then he wins, otherwise, he loses. Nithin doesn't know what he is doing, and he always picks his prime randomly while Antonio always plays optimally. The probability of Antonio winning is $a/b$ where $a$ and $b$ are relatively prime positive integers. Find$a + b$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1998 China Team Selection Test, 1
Find $k \in \mathbb{N}$ such that
[b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots, \left( \begin{array}{c}
n\\
j + k - 1\end{array} \right)$ forms an arithmetic progression.
[b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots , \left( \begin{array}{c}
n\\
j + k - 2\end{array} \right)$ forms an arithmetic progression.
Find all $n$ which satisfies part [b]b.)[/b]
1951 AMC 12/AHSME, 15
The largest number by which the expression $ n^3 \minus{} n$ is divisible for all possible integral values of $ n$, is:
$ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 6$
2016 Indonesia TST, 1
Let $k$ and $n$ be positive integers. Determine the smallest integer $N \ge k$ such that the following holds: If a set of $N$ integers contains a complete residue modulo $k$, then it has a non-empty subset whose sum of elements is divisible by $n$.
2019 India PRMO, 19
If $15$ and $9$ are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer ?
2013 Stanford Mathematics Tournament, 10
Consider a sequence given by $a_n=a_{n-1}+3a_{n-2}+a_{n-3}$, where $a_0=a_1=a_2=1$. What is the remainder of $a_{2013}$ divided by $7$?