Let $ n$ be the smallest positive integer such that $ n$ is divisible by $ 20$, $ n^2$ is a perfect cube, and $ n^3$ is a perfect square. What is the number of digits of $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

$f(x) = x^3-ax+1$ , $a \in R$ has three different zeros in $R$. Prove that for the zero $x_o$ with the smallest absolute value holds: $\frac{1}{a}< x_0 < \frac{2}{a}$

Let $ABC$ be a triangle with $AB=3, BC=4, CA=5$. Let $M$ be the midpoint of $BC$, and $\Gamma$ be a circle through $A$ and $M$ that intersects $AB$ and $AC$ again at $D$ and $E$, respectively. Given that $AD=AE$, find the area of quadrilateral $MEAD$. [i]Individual #9[/i]

Triangle $\vartriangle ABC$ has side lengths $AB = 5$, $BC = 8$, and $CA = 7$. Let the perpendicular bisector of $BC$ intersect the circumcircle of $\vartriangle ABC$ at point $D$ on minor arc $BC$ and point $E$ on minor arc $AC$, and $AC$ at point $F$. The line parallel to $BC$ passing through $F$ intersects $AD$ at point $G$ and $CE$ at point $H$. Compute $\frac{[CHF]}{[DGF]}$ . (Given a triangle $\vartriangle ABC$, $[ABC]$ denotes its area.)

Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

Consider $\vartriangle ABC$ and a point $S$ inside it. Let $A_1, B_1, C_1$ be the intersection points of $AS, BS, CS$ with $BC, AC, AB$, respectively. Prove that at least in one of the resulting quadrilaterals $AB_1SC_1, C_1SA_1B, A_1SB_1C$ both angles at either $C_1$ and $B_1$, or $C_1$ and $A_1$, or $A_1$ and $B_1$ are not acute.

If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, $\cdots$, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None of the preceding} $

A box contains 11 balls, numbered 1,2,3,....,11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? A. $\frac{100}{231}$ B. $\frac{115}{231}$ C. $\frac{1}{2}$ D. $\frac{118}{231}$ E. $\frac{6}{11}$

Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.

Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$. [i]Proposed by Evan Chen[/i]

We say that a positive integer is a [i]good number[/i] if the digit $2$ appears more often than the digit $3$ and that it is a [i]bad number[/i] if the digit $3$ appears more often than the digit $2$. For example, $2023$ is a good number and $123$ is neither good nor bad. Calculate the difference between the quantity of good numbers and the quantity of bad numbers for integers less than or equal to $2023$.

[b](a)[/b] [Problem for class 11] Let r be the inradius and $r_a$, $r_b$, $r_c$ the exradii of a triangle ABC. Prove that $\frac{1}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}$. [b](b)[/b] [Problem for classes 12/13] Let r be the radius of the insphere and let $r_a$, $r_b$, $r_c$, $r_d$ the radii of the four exspheres of a tetrahedron ABCD. (An [i]exsphere[/i] of a tetrahedron is a sphere touching one sideface and the extensions of the three other sidefaces.) Prove that $\frac{2}{r}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}+\frac{1}{r_d}$. I am really sorry for posting these, but else, Orl will probably post them. This time, we really did not have any challenging problem on the DeMO. But at least, the problems were simple enough that I solved all of them. ;) Darij

$P$ is a point in the interior of $\triangle ABC$, and $\angle ABP = \angle PCB = 10^\circ$. (a) If $\angle PBC = 10^\circ$ and $\angle ACP = 20^\circ$, what is the value of $\angle BAP$? (b) If $\angle PBC = 20^\circ$ and $\angle ACP = 10^\circ$, what is the value of $\angle BAP$?

A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $ Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $

Assume that the numbers from the table $$\begin{matrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{matrix}$$satisfy the inequality: $$\sum_{j=1}^n|a_{j1}x_1+a_{j2}x_2+\ldots+a_{jn}x_n|\le M,$$for each choice $x_j=\pm1$. Prove that $$|a_{11}+a_{22}+\ldots+a_{nn}|\le M.$$

Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.

Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $OMG$.

Let $a, b,c$ be positive numbers with $abc = 1$. Prove that $a^3 + b^3 + c^3 + 2[(ab)^3 + (bc)^3 + (ca)^3] \ge 3(a^2b + b^2c + c^2a)$.

Let $n$ be a positive integer number and let $a_1, a_2, \ldots, a_n$ be $n$ positive real numbers. Prove that $f : [0, \infty) \rightarrow \mathbb{R}$, defined by \[f(x) = \dfrac{a_1 + x}{a_2 + x} + \dfrac{a_2 + x}{a_3 + x} + \cdots + \dfrac{a_{n-1} + x}{a_n + x} + \dfrac{a_n + x}{a_1 + x}, \] is a decreasing function. [i]Dan Marinescu et al.[/i]

Let \[ f(t) = \frac{(10 + 9i)t - 10 + 9i}{t + i} , \] where $i=\sqrt{-1}$. Let $P = f(0)$, $Q = f(2023)$, and $R = f(1)$. Determine $\sin^2(m\angle PQR)$.

A point $D$ is taken on the side $BC$ of an acute-angled triangle $ABC$ such that $AB = AD$. Point $E$ on the altitude from $C$ of the triangle is such that the circle $k_1$ with center $E$ is tangent to the line $AD$ at $D$. Let $k_2$ be the circle through $C$ that is tangent to $AB$ at $B$. Prove that $A$ lies on the line determined by the common chord of $k_1$ and $k_2$.

You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair die. If you roll a three, what is the probability that you chose the fair die?

Given a trapezoid with bases $AB$ and $CD$, there exists a point $E$ on $CD$ such that drawing the segments $AE$ and $BE$ partitions the trapezoid into $3$ similar isosceles triangles, each with long side twice the short side. What is the sum of all possible values of $\frac{CD}{AB}$? [i]Proposed by Adam Bertelli[/i]

For positive integer numbers $a$, $b$ and $c$ it is known that $a^2+b^2+c^2$ and $a^3+b^3+c^3$ are both divisible by $a+b+c$. In addition, $gcd(a+b+c, 6) = 1$. Prove that $a^5+b^5+c^5$ is divisible by $(a+b+c)^2$. [i] A. Antropov [/i]

Let $a$, $b$, $c$ be three positive reals such that $(a+b)(b+c)(c+a)=1$. Prove that the following inequality holds: \[ ab+bc+ca \leq \frac 34 . \] [i]Cezar Lupu[/i]