Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$ b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

$x$ is a base-$10$ number such that when the digits of $x$ are interpreted as a base-$20$ number, the resulting number is twice the value as when they are interpreted as a base-$13$ number. Find the sum of all possible values of $x$.

Solve in the reals the following system. $$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$ [i]Gheorghe Marchitan[/i]

Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$. Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$, what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of $k$? [i]2018 CCA Math Bonanza Lightning Round #4.1[/i]

Let $A$ be a real $n\times n$ matrix satisfying $$A+A^{\text T}=I,$$where $A^{\text T}$ denotes the transpose of $A$ and $I$ the $n\times n$ identity matrix. Show that $\det A>0$.

For a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers that are at least $ 1, $ prove that the series $ \sum_{i=1}^{\infty } \frac{1}{x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{1+x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{\lfloor x_i\rfloor } $ converges.

What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

On a right triangle of paper, two points $A$ and $B$ have been painted. You have scissors and you have the right to make cuts (on paper) as follows: cut through a height of the given triangle. In doing so, remove, without the respective altitude, one of the two triangles and continue the process. Prove that after a finite number of cuts you can separate points $A$ and $B$ leaving one of them outside the remaining triangles.

Six people, with distinct weights, want to form a triangular position where there are three people in the bottom row, two in the middle row, and one in the top row, and each person in the top two rows must weigh less than both of their supports. How many distinct formations are there?

Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations \begin{align*}x(y^2+yz+z^2)&=3y+10z\\y(z^2+zx+x^2)&=21z+24x\\z(x^2+xy+y^2)&=7x+28y.\end{align*} Find $xy+yz+zx$.

Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) $ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $

Today is $26$ May $2021$. This date is traditionally in the form ($DD.MM.YYYY$), using $8$ digits, namely $26.05.2021$. Find the nearest day in the future when the traditional date writing will contain $8$ distinct digits.

$2009$ points have been marked on a circle. Lucía colors them with $7$ different colors of her choice. Then Ivan can join three points of the same color, thus forming monochrome triangles. Triangles cannot have points in common; not even vertices in common. Ivan's goal is to draw as many monochrome triangles as possible. Lucía's objective is to prevent Iván's task as much as possible through a good choice of colouring. How many monochrome triangles will Ivan get if they both do their homework to the best of their ability?

For which of the following expressions, there exists an integer $x$ such that the expression is divisble by $25$? $ \textbf{(A)}\ x^3-3x^2+8x-1 \\ \qquad\textbf{(B)}\ x^3+3x^2-2x+1 \\ \qquad\textbf{(C)}\ x^3+14x^2+3x-8 \\ \qquad\textbf{(D)}\ x^3-5x^2+x+1 \\ \qquad\textbf{(E)}\ \text{None of above} $

Let $n\geq 4$ be a positive integer and $x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2}$ be real numbers such that $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$. If there exists an $a>0$ such that \[x_{i}^2=a+x_{i+1}x_{i+2}\quad\forall 1\leq i\leq n\] then prove that at least $2$ of the numbers $x_{1},x_{2},\ldots ,x_{n}$ are negative.

This is a really easy one for Junior level :p $a^2+b^2+c^2+ab+bc+ac=6$ a,b,c>0 Find max{a+b+c}

[b]p16.[/b] When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is $s(t) = t^3 - 3t^2$. His climbing velocity as a function of the strength in his arms is $v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s$. At how many (possibly negative) points in time is Eric stationary? [b]p17[/b]. Consider a triangle $\vartriangle ABC$ with angles $\angle ACB = 60^o$, $\angle ABC = 45^o$. The circumcircle around $\vartriangle ABH$, where $H$ is the orthocenter of $\vartriangle ABC$, intersects $BC$ for a second time in point $P$, and the center of that circumcircle is $O_c$. The line $PH$ intersects $AC$ in point $Q$, and $N$ is center of the circumcircle around $\vartriangle AQP$. Find $\angle NO_cP$. [b]p18.[/b] If $x, y$ are positive real numbers and $xy^3 = \frac{16}{9}$ , what is the minimum possible value of $3x + y$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back? $\textbf{(A)}\ \dfrac{1}{12} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{6} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{2}{3}$

Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$. He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “$+$” sign, and before the other cards he puts a “$-$" sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.

Bread draws a circle. He then selects four random distinct points on the circumference of the circle to form a convex quadrilateral. Kwu comes by and randomly chooses another 3 distinct points (none of which are the same as Bread's four points) on the circle to form a triangle. Find the probability that Kwu's triangle does not intersect Bread's quadrilateral, where two polygons intersect if they have at least one pair of sides intersecting. [i]Proposed by Nathan Cho[/i]

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

The heights $BD$ and $CE$ of the acute-angled triangle $ABC$ intersect at point $H$, the heights of the triangle $ADE$ intersect at point $F$, point $M$ is the midpoint of side $BC$. Prove that $BH + CH \geqslant 2 FM$. [i]A. Kuznetsov[/i]

Find all prime numbers of the form $10101...01$.

A polynomial $f$ with integer coefficients is written on the blackboard. The teacher is a mathematician who has $3$ kids: Andrew, Beth and Charles. Andrew, who is $7$, is the youngest, and Charles is the oldest. When evaluating the polynomial on his kids' ages he obtains: [list]$f(7) = 77$ $f(b) = 85$, where $b$ is Beth's age, $f(c) = 0$, where $c$ is Charles' age.[/list] How old is each child?