Found problems: 85335
Vessel $A$ has liquids $X$ and $Y$ in the ratio $X:Y=8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y=5:9$. What ratio should you mix the liquids in both vessels if you need the mixture to be $X:Y=1:1$?
$\textbf{(A)}~4:3$
$\textbf{(B)}~30:7$
$\textbf{(C)}~17:25$
$\textbf{(D)}~7:30$
Given a triangle $ABC$, incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $M$ be a point inside $ABC$. Prove that $M$ lie on $(I)$ if and only if one number among $\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}$ is sum of two remaining numbers ($S_{ABC}$ denotes the area of triangle $ABC$)
Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.
Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$.
[i]Ray Li.[/i]
A [i] magic square[/i] is a $3\times 3$ grid of numbers in which the sums of the numbers in each row, column, and long diagonal are all equal. How many magic squares exist where each of the integers from $11$ to $19$ inclusive is used exactly once and two of the numbers are already placed as shown below?
$\begin{tabular}{|l|l|l|l|}
\hline
& & 18 \\ \hline
& 15 & \\ \hline
& & \\ \hline
\end{tabular}$
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.
Alexis has $2020$ paintings, the $i$th one of which is a $1\times i$ rectangle for $i = 1, 2, \ldots, 2020$. Compute the smallest integer $n$ for which they can place all of the paintings onto an $n\times n$ mahogany table without overlapping or hanging off the table.
[i]Proposed by Brandon Wang[/i]
In triangle $ABC$ let the $A$-foot be $E$ and the $B$-excenter be $L$. Suppose the incircle of $ABC$ is tangent to $AC$ at $I$. Construct a hyperbola $\mathcal H$ through $A$ with $B$ and $C$ as the foci such that $A$ lies on the branch of the $\mathcal H$ closer to $C$. Construct an ellipse $\mathcal E$ passing through $I$ with $B$ and $C$ as the foci. Suppose $\mathcal E$ meets $\overline{AB}$ again at point $H$. Let $\overline{CH}$ and $\overline{BI}$ intersect the $C$-branch of $\mathcal H$ at points $M$ and $O$ respectively. Prove $E$, $L$, $M$, $O$ are concyclic.
Proposed by [i]Alex Wang[/i]
Show that the equation
$$a^2b=2017(a+b)$$
has no solutions for positive integers $a$ and $b$.
[i]Proposed by Oriol Solé[/i]
The period of a given pendulum on a planet of radius $R$ is constant (unchanged) as we go from the surface of the planet down to radius $a$, where $R > a$. The planet has mass density evenly distributed at any radius $ r < a $. This density is $\rho_0$. Find the total mass of the planet. Express your answer in terms of $\rho_0$, $a$, $R$, the period of the pendulum, $T$, the length of the pendulum string, $L$, and other constants, as necessary.
[b]Warning[/b]: Your answer may contain some math. So be sure to input this correctly!
[i]Problem proposed by Trung Phan[/i]
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
Ada rolls a standard $4$-sided die $5$ times. The probability that the die lands on at most two distinct sides can be written as $ \frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$
A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$.
Show that its area, disregarding sign, is
$$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$
where $\Delta$ is the discriminant of the matrix
$$M=\begin{pmatrix}
a_1 & b_1 &c_1\\
a_2 & b_2 &c_2\\
a_3 & b_3 &c_3
\end{pmatrix}.$$
Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is $a+b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$
Choose arbitrarily $n$ vertices of a regular $2n-$gon and colour them red. The remaining vertices are coloured blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to $17$.
Let $0 < u < 1$ and define
\[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\]
Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.
Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$,
and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.
Show that $$5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}$$
Points $X$, $Y$, and $Z$ lie on a circle with center $O$ such that $XY=12$. Points $A$ and $B$ lie on segment $XY$ such that $OA=AZ=ZB=BO=5$. Compute $AB$.
The equation $ x^3\minus{}ax^2\plus{}bx\minus{}c\equal{}0$ has three (not necessarily different) positive real roots. Find the minimal possible value of $ \frac{1\plus{}a\plus{}b\plus{}c}{3\plus{}2a\plus{}b}\minus{}\frac{c}{b}$.
Given that $P$ is a point inside a tetrahedron with vertices at $A, B, C$ and $D$, such that the sum of the distances $PA+PB+PC+PD$ is a minimum, show that the two angles $\angle APB$ and $\angle CPD$ are equal and are bisected by the same straight line. What other pair of angles must be equal?
Let $O$ be the center of the circle circumscribed around the acute triangle $ABC$, and let $N$ be the midpoint of the arc $ABC$ of this circle. On the sides $AB$ and $BC$ mark points $D$ and $E$ respectively, such that the point $O$ lies on the segment $DE$. The lines $DN$ and $BC$ intersect at the point $P$, and the lines $EN$ and $AB$ intersect at the point $Q$. Prove that $PQ \perp AC$.
Alice and Bob are builders; Charlie is a destroyer. Alice can build a car in $20$ hours and Bob can build a car in $10$ hours, while Charlie destroys a car in $40$ hours. If Alice and Bob are working together on a car Charlie is destroying, how many hours will it take for Alice and Bob to finish building the car?
Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $
[i]Laurențiu Panaitopol[/i]