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]

Let $m,n,k$ be positive integers with $m\ge2$ and $k\ge\log_2(m-1)$. Prove that $$\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.$$

For which values of $n$ it is possible to cover the side wall of staircase of n steps (for $n = 6$ in the figure) with plates of shown shape? The width and height of each step is $1$ dm, the dimensions of plate are $2 \times 2$ dm and from the corner there is cut out a piece with dimensions $1\times 1$ dm. [img]https://cdn.artofproblemsolving.com/attachments/e/e/ac7a52f3dd40480f82024794708c5a449e0c2b.png[/img]

Find all positive integers n with the following property: For every positive divisor $d$ of $n$, $d+1$ divides $n+1$.

Let $\bigtriangleup ABC$ be a triangle and $D$ be a point in line $BC$ such that $AD$ bisects $\angle BAC$. Furthermore, let $F$ and $G$ be points on the circumcircle of $\bigtriangleup ABC$ and $E\neq D$ point in line $BC$ such that $AF=AE=AD=AG$. If $X$ and $Y$ are the feet of perpendiculars from $D$ to $EF$ and $EG,$ respectively. Prove that $XY\parallel AD$.

Find all $n\in \mathbb{N}$, $n>1$ with the following property: All divisors of $n$ can be put in a rectangular table in such way that the sums of the numbers by rows are equal and the sums of the numbers by columns are also equal.

A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$, and $AB+BC=AD+DC$. Let $E$ be the midpoint of arc $BCD$, and $F (\neq C)$ be the antipode of $A$ [i]wrt[/i] $\Gamma$. Let $I,J,K$ be the incenter of $\triangle ABC$, the $A$-excenter of $\triangle ABC$, the incenter of $\triangle BCD$, respectively. Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$. Prove that $EK$ and $PF$ intersect on $\Gamma.$

For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?