Found problems: 85335
1988 Romania Team Selection Test, 11
Let $x,y,z$ be real numbers with $x+y+z=0$. Prove that \[ |\cos x |+ |\cos y| +| \cos z | \geq 1 . \] [i]Viorel Vajaitu, Bogdan Enescu[/i]
2003 National High School Mathematics League, 6
In tetrahedron $ABCD$, $AB=1,CD=3$, the distance between $AB$ and $CD$ is $2$, the intersection angle between $AB$ and $CD$ is $\frac{\pi}{3}$, then the volume of tetrahedron $ABCD$ is
$\text{(A)}\frac{\sqrt3}{2}\qquad\text{(B)}\frac{1}{2}\qquad\text{(C)}\frac{1}{3}\qquad\text{(D)}\frac{\sqrt3}{3}$
2013 Bulgaria National Olympiad, 2
Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying:
$x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$
for all $x,y \in \mathbb{R}$
[i]Proposed by Nikolay Nikolov[/i]
2016 CHKMO, 1
Let $a_1,a_2,\cdots,a_n$ be a sequence of real numbers lying between $1$ and $-1$, i.e. $-1<a_i<1$, for $1\leq i \leq n$ and such that
(i) $a_1+a_2+\cdots+a_n=0$
(ii) $a_1^2+a_2^2+\cdots+a_n^2=40$
Determine the smallest possible value of $n$
2020 LMT Fall, B26
Aidan owns a plot of land that is in the shape of a triangle with side lengths $5$,$10$, and $5\sqrt3$ feet. Aidan wants to plant radishes such that there are no two radishes that are less than $1$ foot apart. Determine the maximum number of radishes Aidan can plant
2017 Saudi Arabia JBMO TST, 6
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
2016 Ecuador Juniors, 4
Two sums, each consisting of $n$ addends , are shown below:
$S = 1 + 2 + 3 + 4 + ...$
$T = 100 + 98 + 96 + 94 +...$ .
For what value of $n$ is it true that $S = T$ ?
1986 AMC 8, 25
Which of the following sets of whole numbers has the largest average?
\[ \textbf{(A)}\ \text{multiples of 2 between 1 and 101} \qquad
\textbf{(B)}\ \text{multiples of 3 between 1 and 101} \\
\textbf{(C)}\ \text{multiples of 4 between 1 and 101} \qquad
\textbf{(D)}\ \text{multiples of 5 between 1 and 101} \\
\textbf{(E)}\ \text{multiples of 6 between 1 and 101}
\]
IV Soros Olympiad 1997 - 98 (Russia), 10.8
Let $a$ be the root of the equation $x^3-x-1=0$. Find an equation of the third degree with integer coefficients whose root is $a^3$.
2008 Canada National Olympiad, 4
Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which
\[ (f(n))^p \equiv n\quad {\rm mod}\; f(p)
\]
for all $ n \in {\bf N}$ and all prime numbers $ p$.
2012 Silk Road, 3
Let $n > 1$ be an integer.
Determine the greatest common divisor of the set of numbers $\left\{ \left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right):0 \le i \le n-1 \right\}$
i.e. the largest positive integer, dividing $\left( \begin{matrix}
2n \\
2i+1 \\
\end{matrix} \right)$ without remainder for every $i = 0, 1, ..., n–1$ .
(Here $\left( \begin{matrix}
m \\
l \\
\end{matrix} \right)=\text{C}_{m}^{l}=\frac{m\text{!}}{l\text{!}\left( m-l \right)\text{!}}$ is binomial coefficient.)
2020 Sharygin Geometry Olympiad, 14
A non-isosceles triangle is given. Prove that one of the circles touching internally its incircle and circumcircle and externally one of its excircles passes through a vertex of the triangle.
Russian TST 2016, P1
A cyclic quadrilateral $ABCD$ is given. Let $I{}$ and $J{}$ be the centers of circles inscribed in the triangles $ABC$ and $ADC$. It turns out that the points $B, I, J, D$ lie on the same circle. Prove that the quadrilateral $ABCD$ is tangential.
2005 Today's Calculation Of Integral, 5
Calculate the following indefinite integrals.
[1] $\int (4-5\tan x)\cos x dx$
[2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$
[3] $\int x^3\sqrt{4-x^2}dx$
[4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$
[5] $\int (3x-4)^2 dx$
2020 USMCA, 2
Sarah is fighting a dragon in DnD. She rolls two fair twenty-sided dice numbered $1, 2, \ldots, 20$. She vanquishes the dragon if the product of her two rolls is a multiple of $4$. What is the probability that the dragon is vanquished?
2021 AMC 10 Spring, 9
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
Kvant 2022, M2688
Let $T_a, T_b$ and $T_c$ be the tangent points of the incircle $\Omega$ of the triangle $ABC$ with the sides $BC, CA$ and $AB{}$ respectively. Let $X, Y$ and $Z{}$ be points on the circle $\Omega$ such that $A{}$ lies on the ray $YX$, $B{}$ lies on the ray $ZY$ and $C{}$ lies on the ray $XZ$. Let $P{}$ be the intersection point of the segments $ZX$ and $T_bT_c$, and similarly $Q=XY \cap T_cT_a$ and $R=YZ\cap T_aT_b$. Prove that the points $A, B$ and $C{}$ lie on the lines $RP, PQ$ and $QR{}$, respectively.
[i]Proposed by L. Shatunov (11th grade student)[/i]
2003 India IMO Training Camp, 9
Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.
2011 USAMTS Problems, 4
A $\emph{luns}$ with vertices $X$ and $Y$ is a region bounded by two circular arcs meeting at the endpoints $X$ and $Y$. Let $A$, $B$, and $V$ be points such that $\angle AVB=75^\circ$, $AV=\sqrt{2}$ and $BV=\sqrt{3}$. Let $\mathcal{L}$ be the largest area luns with vertices $A$ and $B$ that does not intersect the lines $VA$ or $VB$ in any points other than $A$ and $B$. Define $k$ as the area of $\mathcal{L}$. Find the value \[ \dfrac {k}{(1+\sqrt{3})^2}. \]
1988 Romania Team Selection Test, 1
Consider a sphere and a plane $\pi$. For a variable point $M \in \pi$, exterior to the sphere, one considers the circular cone with vertex in $M$ and tangent to the sphere. Find the locus of the centers of all circles which appear as tangent points between the sphere and the cone.
[i]Octavian Stanasila[/i]
2008 Federal Competition For Advanced Students, Part 2, 2
Which positive integers are missing in the sequence $ \left\{a_n\right\}$, with $ a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right]$ for all $ n \ge 1$? ($ \left[x\right]$ denotes the largest integer less than or equal to $ x$, i.e. $ g$ with $ g \le x < g \plus{} 1$.)
2010 AMC 12/AHSME, 2
A big $ L$ is formed as shown. What is its area?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)+fontsize(12pt));
draw((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--cycle);
label("5",(2.5,0),S);
label("2",(5,1),E);
label("2",(1,8),N);
label("8",(0,4),W);[/asy]$ \textbf{(A)}\ 22 \qquad
\textbf{(B)}\ 24 \qquad
\textbf{(C)}\ 26 \qquad
\textbf{(D)}\ 28 \qquad
\textbf{(E)}\ 30$
2009 National Olympiad First Round, 5
What is the perimeter of the right triangle whose exradius of the hypotenuse is $ 30$ ?
$\textbf{(A)}\ 40 \qquad\textbf{(B)}\ 45 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$
1974 AMC 12/AHSME, 19
In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((.82,0)--(1,1)--(0,.76)--cycle);
label("A", (0,0), S);
label("B", (1,0), S);
label("C", (1,1), N);
label("D", (0,1), N);
label("M", (0,.76), W);
label("N", (.82,0), S);
[/asy]
$ \textbf{(A)}\ 2\sqrt{3}-3 \qquad\textbf{(B)}\ 1-\frac{\sqrt{3}}{3} \qquad\textbf{(C)}\ \frac{\sqrt{3}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{3} \qquad\textbf{(E)}\ 4-2\sqrt{3} $
2003 USA Team Selection Test, 5
Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.