Found problems: 85335
Let be a twice-differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the properties that:
$ \text{(i) supp} f''=f\left(\mathbb{R}\right) $
$ \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R}\quad\forall x\in\mathbb{R}\quad f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right) $
Prove that:
[b]a)[/b] any such $ g $ is injective.
[b]b)[/b] $ f $ is of class $ C^{\infty } , $ and for any natural number $ n, $ any real number $ x $ and any such $ g, $
$$f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) . $$
[i]Laurențiu Panaitopol[/i]
If $ABCDEF$ is a convex cyclic hexagon, then its diagonals $AD$, $BE$, $CF$ are concurrent if and only if $\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$.
[i]Alternative version.[/i] Let $ABCDEF$ be a hexagon inscribed in a circle. Then, the lines $AD$, $BE$, $CF$ are concurrent if and only if $AB\cdot CD\cdot EF=BC\cdot DE\cdot FA$.
Find all positive integers $m$ and $n$ for which \[1!+2!+3!+\cdots+n!=m^{2}\]
A straight line $\ell$ and a point $A$ outside of it are given on the plane. Find the locus of the vertices $C$ of the equilateral triangle $ABC$, the vertex $B$ of which lies on the straight line $\ell$.
Let be a natural number $ n, $ a positive real number $ \lambda , $ and a complex number $ z. $ Prove the following inequalities.
$$ 0\le -\lambda +\frac{1}{n}\sum_{\stackrel{w\in\mathbb{C}}{w^n=1 }} \left| z-\lambda w \right|\le |z| $$
[i]Gheorghe Iurea[/i]
Let $m,n$ are positive integers.
a)Prove that $(m,n)=2\sum_{k=0}^{m-1}[\frac{kn}{m}]+m+n-mn$.
b)If $m,n\geq 2$, prove that $\sum_{k=0}^{m-1}[\frac{kn}{m}]=\sum_{k=0}^{n-1}[\frac{km}{n}]$.
On the coordinate plane, let $C$ be a circle centered $P(0,\ 1)$ with radius 1. let $a$ be a real number $a$ satisfying $0<a<1$. Denote by $Q,\ R$ intersection points of the line $y=a(x+1) $ and $C$.
(1) Find the area $S(a)$ of $\triangle{PQR}$.
(2) When $a$ moves in the range of $0<a<1$, find the value of $a$ for which $S(a)$ is maximized.
[i]2011 Tokyo University entrance exam/Science, Problem 1[/i]
$f(n,k)$ is defined by
(1) $f(n,0) = f(n,n) = 1$ and
(2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$.
How many times do we need to use (2) to find $f(3991,1993)$?
Find all possible values of the number
$$\frac{a + b}{c}+\frac{a + c}{b}+\frac{b + c}{a},$$
where $a, b, c$ are positive integers, and $\frac{a + b}{c},\frac{a + c}{b},\frac{b + c}{a}$ are also positive integers.
Find the positive integers $x_1, x_2, \ldots, x_{29}$ at least one of which is greater that 1988 so that
\[ x^2_1 + x^2_2 + \ldots x^2_{29} = 29 \cdot x_1 \cdot x_2 \ldots x_{29}. \]
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of
\[
\sum_{i = 1}^{2012} | a_i - i |,
\]
then compute the sum of the prime factors of $S$.
[i]Proposed by Aaron Lin[/i]
How does one show $$\text{lcm}\left(\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\right)=\frac{\text{lcm}(1,2,\ldots,n+1)}{n+1}$$
Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.
Find all triples of natural numbers $(a,b,c)$ such that $a$, $b$ and $c$ are in geometric progression and $a + b + c = 111$.
Given $(2m+1)$ different integers, each absolute value is not greater than $(2m-1)$. Prove that it is possible to choose three numbers among them, with their sum equal to zero.
Let $a>1$ and $c$ be natural numbers and let $b\neq 0$ be an integer. Prove that there exists a natural number $n$ such that the number $a^n+b$ has a divisor of the form $cx+1$, $x\in\mathbb{N}$.
Let $a$ and $b$ be the solutions to $x^2-x+q=0$, find
$a^3+b^3+3(a^3b+ab^3)+6(a^3b^2+a^2b^3)$.
We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday the number we record is ten and a half. On every Tuesday and every Thursday the number we record is one third of what it was on the previous workday. On every Monday, Wednesday, and Friday the number we record is double what it was on the previous workday. This will go on forever. What is the sum of all of the numbers we will record?
There are lamps in every field of $n\times n$ table. At start all the lamps are off. A move consists of chosing $m$ consecutive fields in a row or a column and changing the status of that $m$ lamps. Prove that you can reach a state in which all the lamps are on only if $m$ divides $n.$
There are $212$ points inside or on a given unit circle. Prove that there are at least $2001$ pairs of points having distances at most $1$.
A square $ (n \minus{} 1) \times (n \minus{} 1)$ is divided into $ (n \minus{} 1)^2$ unit squares in the usual manner. Each of the $ n^2$ vertices of these squares is to be coloured red or blue. Find the number of different colourings such that each unit square has exactly two red vertices. (Two colouring schemse are regarded as different if at least one vertex is coloured differently in the two schemes.)
[u]Part 1[/u]
You might think this round is broken after solving some of these problems, but everything is intentional.
[b]1.1.[/b] The number $n$ can be represented uniquely as the sum of $6$ distinct positive integers. Find $n$.
[b]1.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = 5$ and $AC^2 = 1188$. Find $AB$.
[b]1.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a \$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $15 + 5k$ cents. The amount it made in sales for the previous month was $\$40.50$. Assuming the stand spent its entire budget on production and advertising, what was the absolute di erence between the amount spent on production and the amount spent on advertising?
[b]1.4.[/b] Let $A$ be the number of di erent ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 1$, $1 \times 3$, and $1 \times 6$. Let B be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 2$ and $1 \times 5$, where there are 2 different colors available for the $1 \times 2$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.)
[b]1.5.[/b] An integer $n \ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$.
[b]1.6.[/b] Let $x$ be a positive integer such that $3$, $ \log_6(12x)$, $\log_6(18x)$ form an arithmetic progression in some order. Find $x$.
[u]Part 2[/u]
Oops, it looks like there were some [i]intentional [/i] printing errors and some of the numbers from these problems got removed. Any $\blacksquare$ that you see was originally some positive integer, but now its value is no longer readable. Still, if things behave like they did for Part 1, maybe you can piece the answers together.
[b]2.1.[/b] The number $n$ can be represented uniquely as the sum of $\blacksquare$ distinct positive integers. Find $n$.
[b]2.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = \blacksquare$ and $AC^2 = 1536$. Find $AB$.
[b]2.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a $\$50$ dollar budget to divide between production and advertising. If it spent k dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $25 + 5k$ cents. The amount it made in sales for the previous month was $\$\blacksquare$. Assuming the stand spent its entire budget on production and advertising, what was the absolute di erence between the amount spent on production and the amount spent on advertising?
[b]2.4.[/b] Let $A$ be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times \blacksquare$, $1 \times \blacksquare$, and $1 \times \blacksquare$. Let $B$ be the number of different ways to tile a $1\times n$ rectangle with tiles of size $1 \times \blacksquare$ and $1 \times \blacksquare$, where there are $\blacksquare$ different colors available for the $1 \times \blacksquare$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.)
[b]2.5.[/b] An integer $n \ge \blacksquare$ is such that $n$ when represented in base $9$ is written the same way as $2n$ is in base $\blacksquare$. Find $n$.
[b]2.6.[/b] Let $x$ be a positive integer such that $1$, $\log_{96}(6x)$, $\log_{96}(\blacksquare x)$ form an arithmetic progression in some order. Find $x$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
For $ p\equal{}1,2,\ldots,10$ let $ S_p$ be the sum of the first $ 40$ terms of the arithmetic progression whose first term is $ p$ and whose common difference is $ 2p\minus{}1$; then $ S_1\plus{}S_2\plus{}\cdots\plus{}S_{10}$ is
$ \textbf{(A)}\ 80000
\qquad \textbf{(B)}\ 80200
\qquad \textbf{(C)}\ 80400
\qquad \textbf{(D)}\ 80600
\qquad \textbf{(E)}\ 80800$
Let $AD$ be the perpendicular to the hypotenuse $BC$ of the right triangle $ABC$. Let $DE$ be the height of the triangle $ADB$ and $DZ$ be the height of the triangle $ADC$. On the line $AB$ is chosen the point $N$ so that $CN$ is parallel to $EZ$. Let $A'$ be symmetrical of $A$ to $EZ$ and $I, K$ projections of $A'$ on $AB$, respectively, on $AC$. Prove that $<$ $NA'T$ $=$ $<$ $ADT$, where $T$ is the point of intersection of $IK$ and $DE$.
If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that
$\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$
is also a rational number.