Found problems: 85335
1981 Spain Mathematical Olympiad, 5
Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this:
$$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$
a) Represent the function graphically.
b) Calculate $A_n =\int_0^1 f_n(x) dx$.
c) Find, if it exists, $\lim_{n\to \infty} A_n$ .
2021 BMT, 1
The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$, while its charge is $\frac12$ at pH $9.6$. Charge increases linearly with pH. What is the isoelectric point of glycine?
2011 Today's Calculation Of Integral, 761
Find $\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.$
1975 Czech and Slovak Olympiad III A, 1
Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)
2013 HMNT, 10
Let $\omega= \cos \frac{2\pi}{727} + i \sin \frac{2\pi}{727}$. The imaginary part of the complex number
$$\prod^{13}_{k=8} \left(1 + \omega^{3^{k-1}}+ \omega^{2\cdot 3^{k-1}}\right)$$ is equal to $\sin a$ for some angle $a$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ , inclusive. Find $a$.
2021 AMC 10 Fall, 5
Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$?
$\textbf{(A) }4^{1010}\qquad\textbf{(B) }2^{2022}\qquad\textbf{(C) }8^{2018}\qquad\textbf{(D) }4^{3031}\qquad\textbf{(E) }4^{3032}$
2018 Ramnicean Hope, 3
[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism.
[b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $
[b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $
[i]Marcel Țena[/i]
1981 All Soviet Union Mathematical Olympiad, 308
Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$
2023 Bulgaria JBMO TST, 4
Given is an acute angled triangle $ABC$ with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$ and $P$ be the point of intersection of $\omega$ and $k$ other than $A$. Assume that $BP$ and $CP$ intersect $\omega$ for the second time at points $Q$ and $R$, respectively. If $D$ is the foot of the altitude from $A$ to $BC$ and $S$ is the point of the intersection of $\omega$ and $QD$, prove that $HR = HS$.
2011 ELMO Problems, 4
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
2021 China Second Round Olympiad, Problem 6
A sequence $\{a_n\}$ satisfies $$a_0=0, a_1=a_2=1, a_{3n} = a_n, a_{3n+1}=a_{3n+2} = a_n+1$$ for all $n \geq 1$. Compute $a_{2021}$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 6)[/i]
2023 Brazil Team Selection Test, 1
Let $n \geq 5$ be an integer. Consider $n$ squares with side lengths $1, 2, \dots , n$, respectively. The squares are arranged in the plane with their sides parallel to the $x$ and $y$ axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.
2018 Romania National Olympiad, 3
Let $f,g : \mathbb{R} \to \mathbb{R}$ be two quadratics such that, for any real number $r,$ if $f(r)$ is an integer, then $g(r)$ is also an integer.
Prove that there are two integers $m$ and $n$ such that $$g(x)=mf(x)+n, \: \forall x \in \mathbb{R}$$
2000 USAMO, 4
Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
1968 All Soviet Union Mathematical Olympiad, 094
Given an octagon with the equal angles. The lengths of all the sides are integers. Prove that the opposite sides are equal in pairs.
[u]alternate wording[/u]
Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.
2018 CHMMC (Fall), 5
Let $\vartriangle ABC$ be a right triangle such that $AB = 3$, $BC = 4$, $AC = 5$. Let point $D$ be on $AC$ such that the incircles of $\vartriangle ABD$ and $\vartriangle BCD$ are mutually tangent. Find the length of $BD$.
2006 Stanford Mathematics Tournament, 5
There exist two positive numbers $ x$ such that $ \sin(\arccos(\tan(\arcsin x)))\equal{}x$. Find the product of the two possible $ x$.
2016 ASDAN Math Tournament, 2
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?
1995 IMO Shortlist, 4
An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that
\[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\]
The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
1977 Bulgaria National Olympiad, Problem 3
A given truncated pyramid has triangular bases. The areas of the bases are $B_1$ and $B_2$ and the area of the surface is $S$. Prove that if there exists a plane parallel to the bases whose intersection divides the pyramid to two truncated pyramids in which may be inscribed by spheres then
$$S=(\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2$$
[i]G. Gantchev[/i]
2007 Princeton University Math Competition, 1
A pirate ship spots, $10$ nautical miles to the east, an oblivious caravel sailing $60$ south of west at a steady $12 \text{ nm/hour}$. What is the minimum speed that the pirate ship must maintain at to be able to catch the caravel?
II Soros Olympiad 1995 - 96 (Russia), 10.6
On sides $BC$, $CA$ and $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are taken, respectively, so that the radii of the circles inscribed in triangles $A_1BC_1$, $AB_1C_1$ and $A_1B_1C$ are equal to each other and equal to $r$. The radius of the circle inscribed in triangle $A_1B_1C_1$ is equal to $r_1$. Find the radius of the circle inscribed in triangle $ABC$.
2023 CMIMC Team, 2
Real numbers $x$ and $y$ satisfy
\begin{align*}
x^2 + y^2 &= 2023 \\
(x-2)(y-2) &= 3.
\end{align*}
Find the largest possible value of $|x-y|$.
[i]Proposed by Howard Halim[/i]
1990 Putnam, B6
Let $S$ be a nonempty closed bounded convex set in the plane. Let $K$ be a line and $t$ a positive number. Let $L_1$ and $L_2$ be support lines for $S$ parallel to $K_1$, and let $ \overline {L} $ be the line parallel to $K$ and midway between $L_1$ and $L_2$. Let $B_S(K,t)$ be the band of points whose distance from $\overline{L}$ is at most $ \left( \frac {t}{2} \right) w $, where $w$ is the distance between $L_1$ and $L_2$. What is the smallest $t$ such that \[ S \cap \bigcap_K B_S (K, t) \ne \emptyset \]for all $S$? ($K$ runs over all lines in the plane.)
2022 Math Prize for Girls Problems, 13
The roots of the polynomial $x^4 - 4ix^3 +3x^2 -14ix - 44$ form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?