Found problems: 85335
1997 National High School Mathematics League, 5
Let $f(x)=x^2-\pi x$, $\alpha=\arcsin\frac{1}{3},\beta=\arctan\frac{5}{4},\gamma=\arccos\left(-\frac{1}{3}\right),\delta=\text{arccot}\left(-\frac{5}{4}\right)$
$\text{(A)}f(\alpha)>f(\beta)>f(\delta)>f(\gamma)$
$\text{(B)}f(\alpha)>f(\delta)>f(\beta)>f(\gamma)$
$\text{(C)}f(\delta)>f(\alpha)>f(\beta)>f(\gamma)$
$\text{(D)}f(\delta)>f(\alpha)>f(\gamma)>f(\beta)$
2019 BMT Spring, 5
Find the area of the set of all points $ z $ in the complex plane that satisfy $ \left| z - 3i \right| + \left| z - 4 \right| \leq 5\sqrt{2} $.
2015 Paraguayan Mathematical Olympiad, Problem 1
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
2024 IFYM, Sozopol, 1
Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) such that:
\[
f(x^2 + y) = xf(x) + \frac{f(y^2)}{y}
\]
for any positive real numbers \( x \) and \( y \).
ICMC 2, 2
This question, again, comprises two independent parts.
(i) Show that if \((k+1)\) integers are chosen from \(\left\{1,2,3,...,2k+1\right\}\), then among the chosen integers there are always two that are coprime.
(ii) Let \(A=\left\{1,2,\ldots,n\right\}.\) Prove that if \(n>11\) then there is a bijective map \(f: A\to A\) with the property that, for every \(a\in A\), exactly one of \(f(f(f(f(a))))=a\) and \(f(f(f(f(f(a)))))=a\) holds.
1982 AMC 12/AHSME, 4
The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is
$\text{(A)} \pi \qquad \text{(B)} \frac{2}{\pi} \qquad \text{(C)} 1 \qquad \text{(D)} \frac{1}{2} \qquad \text{(E)} \frac{4}{\pi} + 2$
1979 Chisinau City MO, 174
Prove that for any odd number $a$ there exists an integer $b$ such that $2^b-1$ is divisible by $a$.
2014 Harvard-MIT Mathematics Tournament, 25
Let $ABC$ be an equilateral triangle of side length $6$ inscribed in a circle $\omega$. Let $A_1,A_2$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_1,B_2,C_1,C_2$ similarly. Given that $A_1,A_2,B_1,B_2,C_1,C_2$ appear on $\omega$ in that order, find the area of hexagon $A_1A_2B_1B_2C_1C_2$.
2019 LIMIT Category B, Problem 11
Let $S=\{1,2,\ldots,10\}$. Three numbers are chosen with replacement from $S$. If the chosen numbers denote the lengths of sides of a triangle, then the probability that they will form a triangle is:
$\textbf{(A)}~\frac{101}{200}$
$\textbf{(B)}~\frac{99}{200}$
$\textbf{(C)}~\frac12$
$\textbf{(D)}~\frac{110}{200}$
2019 Macedonia Junior BMO TST, 2
Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$. Let $t_{1}$ and $t_{2}$ be the tangents to $\omega_{1}$ and $\omega_{2}$, respectively, at point $A$. Let the second intersection of $\omega_{1}$ and $t_{2}$ be $C$, and let the second intersection of $\omega_{2}$ and $t_{1}$ be $D$. Points $P$ and $E$ lie on the ray $AB$, such that $B$ lies between $A$ and $P$, $P$ lies between $A$ and $E$, and $AE = 2 \cdot AP$. The circumcircle to $\bigtriangleup BCE$ intersects $t_{2}$ again at point $Q$, whereas the circumcircle to $\bigtriangleup BDE$ intersects $t_{1}$ again at point $R$. Prove that points $P$, $Q$, and $R$ are collinear.
2016 India PRMO, 11
For real numbers $x$ and $y$, let $M$ be the maximum value of the expression $x^4y + x^3y + x^2y + xy + xy^2 + xy^3 + xy^4$, subject to $x + y = 3$. Find $[M]$.
2021 Brazil Undergrad MO, Problem 3
Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.
1951 Moscow Mathematical Olympiad, 200
What figure can the central projection of a triangle be? (The center of the projection does not lie on the plane of the triangle.)
2017 Lusophon Mathematical Olympiad, 5
The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.
1980 IMO Shortlist, 14
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
2019 AMC 10, 20
As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
\[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\]
where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$?
[asy]
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filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0));
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label("$A$",(-3,-1),S);
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label("$E$",(-2,0),NW);
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label("$B$",(-1,-1),S);
dot((0,0));
label("$F$",(0,0),N);
dot((1,-1));
label("$C$",(1,-1), S);
dot((2,0));
label("$G$", (2,0),NE);
dot((3,-1));
label("$D$", (3,-1), S);
[/asy]
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$
2019 LIMIT Category B, Problem 7
Find the number of ordered pairs of positive integers for which
$$\frac1a+\frac1b=\frac4{2019}$$
2012 France Team Selection Test, 1
Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$).
1) If $k=2n+1$, prove that there exists a person who knows all others.
2) If $k=2n+2$, give an example of such a group in which no-one knows all others.
1977 USAMO, 3
If $ a$ and $ b$ are two of the roots of $ x^4\plus{}x^3\minus{}1\equal{}0$, prove that $ ab$ is a root of $ x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$.
1968 Polish MO Finals, 1
What is the largest number of regions into which a plane can be divided by drawing $n$ pairs of parallel lines?
2021 May Olympiad, 1
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.
2011 Iran MO (3rd Round), 4
A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter.
[i]proposed by Masoud Nourbakhsh[/i]
2009 Regional Olympiad of Mexico Center Zone, 3
An equilateral triangle $ABC$ has sides of length $n$, a positive integer. Divide the triangle into equilateral triangles of length $ 1$, drawing parallel lines (at distance $ 1$) to all sides of the triangle. A path is a continuous path, starting at the triangle with vertex $A$ and always crossing from one small triangle to another on the side that both triangles share, in such a way that it never passes through a small triangle twice. Find the maximum number of triangles that can be visited.
2020 Ukrainian Geometry Olympiad - December, 3
On the sides $AB$ and $AC$ of a triangle $ABC$ select points $D$ and $E$ respectively, such that $AB = 6$, $AC = 9$, $AD = 4$ and $AE = 6$. It is known that the circumscribed circle of $\vartriangle ADE$ interects the side $BC$ at points $F, G$ , where $BF < BG$. Knowing that the point of intersection of lines $DF$ and $EG$ lies on the circumscribed circle of $\vartriangle ABC$ , find the ratio $BC:FG$.
2015 CIIM, Problem 6
Show that there exists a real $C > 1$ that satisfy the following property: if $n > 1$ and $a_0 < a_1 < \cdots < a_n$ are positive integers such that $\frac{1}{a_0},\frac{1}{a_1},\dots,\frac{1}{a_n}$ are in arithmetic progression, then $a_0 > C^n.$