Found problems: 85335
2010 Kazakhstan National Olympiad, 3
Positive real $A$ is given. Find maximum value of $M$ for which inequality
$ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $
holds for all $x, y>0$
2012 Philippine MO, 2
Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.
Putnam 1939, B2
Evaluate $\int_{1}^{3} ( (x - 1)(3 - x) )^{\dfrac{-1}{2}} dx$ and $\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} dx.$
2007 Mediterranean Mathematics Olympiad, 2
The diagonals $AC$ and $BD$ of a convex cyclic quadrilateral $ABCD$ intersect at point $E$. Given that $AB = 39, AE = 45, AD = 60$ and $BC = 56$, determine the length of $CD.$
2016 Belarus Team Selection Test, 1
a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url])
b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$
Kvant 2024, M2816
Find out for which natural numbers $m$ it is possible to find a natural $\ell$ such that the sum of $n+n^2+n^3+\ldots+n^\ell$ will be divisible by $m$ for any natural $n$.
[i]A. Skabelin[/i]
2012 Indonesia TST, 2
The positive integers are colored with black and white such that:
- There exists a bijection from the black numbers to the white numbers,
- The sum of three black numbers is a black number, and
- The sum of three white numbers is a white number.
Find the number of possible colorings that satisfies the above conditions.
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?