Found problems: 85335
1985 Miklós Schweitzer, 9
Let $D=\{ z\in \mathbb C\colon |z|<1\}$ and $D=\{ w\in \mathbb C \colon |w|=1\}$. Prove that if for a function $f\colon D\times B\rightarrow\mathbb C$ the equality
$$f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)$$
holds for all $z\in D, w\in B$ and $a, b\in \mathbb C,|a|^2=|b|^2+1$, then there is a function $L\colon (0, \infty )\rightarrow \mathbb C$ satisfying
$$L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0$$
such that $f$ can be represented as
$$f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B$$.
[Gy. Maksa]
2003 Cuba MO, 9
Let $D$ be the midpoint of the base $AB$ of the isosceles and acute angle triangle $ABC$, $E$ is a point on $AB$ and $O$ circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ perpendicular to $DO$, the line that passes through $E$ perpendicular to $BC$ and the line that passes through$ B$ parallel to $AC$, they intersect at a point.
2022 Bosnia and Herzegovina IMO TST, 4
In each square of a $4 \times 4$ table a number $0$ or $1$ is written, such that the product of every two neighboring squares is $0$ (neighboring by side).
$a)$ In how many ways is this possible to do if the middle $2\times 2$ is filled with $4$ zeros?
$b)$ In general, in how many ways is this possible to do (regardless of the middle $2 \times 2$)?
2008 CentroAmerican, 4
Five girls have a little store that opens from Monday through Friday. Since two people are always enough for taking care of it, they decide to do a work plan for the week, specifying who will work each day, and fulfilling the following conditions:
a) Each girl will work exactly two days a week
b) The 5 assigned couples for the week must be different
In how many ways can the girls do the work plan?
2017 Mediterranean Mathematics Olympiad, Problem 4
Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that
$$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$
LMT Team Rounds 2010-20, 2020.S29
Let $\mathcal{F}$ be the set of polynomials $f(x)$ with integer coefficients for which there exists an integer root of the equation $f(x)=1$. For all $k>1$, let $m_k$ be the smallest integer greater than one for which there exists $f(x)\in \mathcal{F}$ such that $f(x)=m_k$ has exactly $k$ distinct integer roots. If the value of $\sqrt{m_{2021}-m_{2020}}$ can be written as $m\sqrt{n}$ for positive integers $m,n$ where $n$ is squarefree, compute the largest integer value of $k$ such that $2^k$ divides $\frac{m}{n}$.
1997 Chile National Olympiad, 4
The [i]triangular domino[/i] is a game that uses the tokens shown below, with equilateral triangle shape with side $ 1$. The idea of the game is to construct an equilateral triangle with side $n$, no gaps, following the rules of the domino or classic.
$\bullet$ Show that the sum $S$ of the values corresponding to the edges that are part of the sides of the greater triangle, it depends only on n, and not on the way in which the tokens are paired.
$\bullet$ For each value of $n$, calculate $S$.
[img]https://cdn.artofproblemsolving.com/attachments/e/9/898664fac380725a7398dfe470298a90b8c69b.png[/img]
2021 Macedonian Team Selection Test, Problem 3
A group of people is said to be [i]good[/i] if every member has an even number (zero included) of acquaintances in it. Prove that any group of people can be partitioned into two (possibly empty) parts such that each part is good.
2017 ISI Entrance Examination, 3
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function given by
$$f(x) =\begin{cases} 1 & \mbox{if} \ x=1 \\ e^{(x^{10}-1)}+(x-1)^2\sin\frac1{x-1} & \mbox{if} \ x\neq 1\end{cases}$$
(a) Find $f'(1)$
(b) Evaluate $\displaystyle \lim_{u\to\infty} \left[100u-u\sum_{k=1}^{100} f\left(1+\frac{k}{u}\right)\right]$.
2008 Harvard-MIT Mathematics Tournament, 20
For how many ordered triples $ (a,b,c)$ of positive integers are the equations $ abc\plus{}9 \equal{} ab\plus{}bc\plus{}ca$ and $ a\plus{}b\plus{}c \equal{} 10$ satisfied?
2021 USMCA, 6
Let $ABCD$ be a unit square. Construct point $E$ outside $ABCD$ such that $\overline{AE} = \sqrt{2} \cdot \overline{BE}$ and $\angle{AEB} = 135^{\circ}$. Also, let $F$ be the foot of the perpendicular from $A$ to line $BE$. Find the area of $\triangle{BDF}$.
2016 South East Mathematical Olympiad, 7
Let $A=\{a^3+b^3+c^3-3abc|a,b,c\in\mathbb{N}\}$, $B=\{(a+b-c)(b+c-a)(c+a-b)|a,b,c\in\mathbb{N}\}$, $P=\{n|n\in A\cap B,1\le n\le 2016\}$, find the value of $|P|$.
1939 Moscow Mathematical Olympiad, 043
Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\
x + y+ z = 2b \\
x^2 + y^2-z^2 = b^2
\end{cases}$ in $C$
1986 Vietnam National Olympiad, 3
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.
2022 Moldova Team Selection Test, 10
Let $P(X)$ be a polynomial with positive coefficients. Show that for every integer $n \geq 2$ and every $n$ positive numbers $x_1, x_2,..., x_n$ the following inequality is true: $$P\left(\frac{x_1}{x_2} \right)^2+P\left(\frac{x_2}{x_3} \right)^2+ ... +P\left(\frac{x_n}{x_1} \right)^2 \geq n \cdot P(1)^2.$$ When does the equality take place?
2017 Purple Comet Problems, 17
The expression $\left(1 + \sqrt[6]{26 + 15\sqrt3} -\sqrt[6]{26 - 15\sqrt3}\right)^6= m + n\sqrt{p}$ , where $m, n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.
2002 Italy TST, 2
Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides
\[\binom{p^n}{p}-p^{n-1}. \]
2020/2021 Tournament of Towns, P7
A white bug sits in one corner square of a $1000$ × $n$ chessboard, where $n$ is an odd positive integer and $n > 2020$. In the two nearest corner squares there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the opposite corner square by never visiting a square occupied or attacked by a bishop, and visiting every other square exactly once. Show that the number of ways for the bug to attain its goal does not depend on $n$.
2009 IMAC Arhimede, 2
In the triangle $ABC$, the circle with the center at the point $O$ touches the pages $AB, BC$ and $CA$ in the points $C_1, A_1$ and $B_1$, respectively. Lines $AO, BO$ and $CO$ cut the inscribed circle at points $A_2, B_2$ and $C_2,$ respectively. Prove that it is the area of the triangle $A_2B_2C_2$ is double from the surface of the hexagon $B_1A_2C_1B_2A_1C_2$.
(Moldova)
2018 Online Math Open Problems, 2
Let $(p_1, p_2, \dots) = (2, 3, \dots)$ be the list of all prime numbers, and $(c_1, c_2, \dots) = (4, 6, \dots)$ be the list of all composite numbers, both in increasing order. Compute the sum of all positive integers $n$ such that $|p_n - c_n| < 3$.
[i]Proposed by Brandon Wang[/i]
1992 AMC 8, 4
During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?
$\text{(A)}\ 28\% \qquad \text{(B)}\ 35\% \qquad \text{(C)}\ 70\% \qquad \text{(D)}\ 75\% \qquad \text{(E)}\ 80\% $
2008 Germany Team Selection Test, 2
Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.
2009 Putnam, B2
A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$
2016 Purple Comet Problems, 7
Positive integers m and n are both greater 50, have a least common multiple equal to 480, and have a
greatest common divisor equal to 12. Find $m + n$.
1955 Czech and Slovak Olympiad III A, 4
Given that $a,b,c$ are distinct real numbers, show that the equation
\[\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x-c}=0\]
has a real root.