Let $n \ge 3$ and $\sigma \in S_n$ a permutation of the first $n$ positive integers. Prove that the numbers $\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)$ cannot form an arithmetic, nor a geometric progression.

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $ 40 \%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $ 30 \%$ of the group are girls. How many girls were initially in the group? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

Two regular pentagons $ABCDE$ and $AEKPL$ are given in space, such that $\angle DAK = 60^{\circ}$. Let $M$, $N$ and $S$ be the midpoints of $AE$, $CD$ and $EK$. Prove that: [list=a] [*] $\triangle NMS$ is a right triangle; [*] planes $(ACK)$ and $(BAL)$ are perpendicular. [/list] [i]Ukraine Olympiad[/i]

The diagram shows a magic square in which the sums of the numbers in any row, column or diagonal are equal. What is the value of $n$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 11$

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. If $O_a, O_b, O_c$ denote the circumcenters of $\triangle AOH$, $\triangle BOH$, $\triangle COH$, then prove that $AO_a, BO_b, CO_c$ are concurrent.

There are$n$ balls numbered $1,2,\cdots,n$, respectively. They are painted with $4$ colours, red, yellow, blue, and green, according to the following rules: First, randomly line them on a circle. Then let any three clockwise consecutive balls numbered $i, j, k$, in order. 1) If $i>j>k$, then the ball $j$ is painted in red; 2) If $i<j<k$, then the ball $j$ is painted in yellow; 3) If $i<j, k<j$, then the ball $j$ is painted in blue; 4) If $i>j, k>j$, then the ball $j$ is painted in green. And now each permutation of the balls determine a painting method. We call two painting methods distinct, if there exists a ball, which is painted with two different colours in that two methods. Find out the number of all distinct painting methods.

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

What is the largest power of 2 that is a divisor of $13^4-11^4$? $\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 32\qquad\textbf{(D) } 64\qquad \textbf{(E) } 128$

A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$. [asy] draw((30,0)--(-70,0), Arrow); draw((30,0)--(-20,-40)); draw((-20,-40)--(80,-40), Arrow); draw((0,-60)--(-40,20), dashed, Arrow); draw((0,-60)--(0,15), dashed); draw((0,15)--(40,-65),dashed, Arrow); [/asy]

What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other? $\text{(A) }\frac{1}{10}\qquad\text{(B) }\frac{2}{15}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{3}{10}\qquad\text{(E) }\frac{2}{5}$

Acute triangle $ABC$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $BC.$ Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM.$ Segments $PQ$ and $BC$ intersect at $S$. Suppose that $BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}},$ and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^2$ can be expressed as $\tfrac ab$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.

Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called [i]good[/i] if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called [i]very good[/i] if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of [i]good[/i] elements in $M$ and $v$ denote the number of [i]very good[/i] elements in $M.$ Prove that \[v^2+v \leq g \leq n^2-n.\]

Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called [i]discrete coloring[/i] if there exists a circle which encloses all red points and excludes all white points. Determine the number of [i]discrete colorings[/i] for each set $S$.

Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as $x\pi$ for some value of $x$. Compute $x$. [i]2022 CCA Math Bonanza Team Round #3[/i]

Let $ABC$ be a triangle such that $AB=9$, $BC=6$, and $AC=10$. $2$ points $D_1,D_2$ are labeled on $BC$ such that $BC$ is subdivided into $3$ equal segments; $4$ points $E_1,E_2,\dots,E_4$ are labeled on $AC$ such that $AC$ is subdivided into $5$ equal segments; and $8$ points $F_1,F_2,\dots,F_8$ are labeled on $AB$ such that $AB$ is subdivided into $9$ equal segments. All possible cevians are drawn from $A$ to each $D_i$; from $B$ to each $E_j$; and from $C$ to each $F_k$. At how many points in the interior of $\triangle ABC$ do at least $2$ cevians intersect?

For every permutation $ \tau$ of $ 1, 2, \ldots, 10$, $ \tau \equal{} (x_1, x_2, \ldots, x_{10})$, define $ S(\tau) \equal{} \sum_{k \equal{} 1}^{10} |2x_k \minus{} 3x_{k \minus{} 1}|$. Let $ x_{11} \equal{} x_1$. Find [b]I.[/b] The maximum and minimum values of $ S(\tau)$. [b]II.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its maximum. [b]III.[/b] The number of $ \tau$ which lets $ S(\tau)$ attain its minimum.

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

For a positive integer $n$, denote by $\tau (n)$ the number of its positive divisors. For a positive integer $n$, if $\tau (m) < \tau (n)$ for all $m < n$, we call $n$ a good number. Prove that for any positive integer $k$, there are only finitely many good numbers not divisible by $k$.

The streets of a town are arranged in three directions , dividing the town into blocks which are equilateral triangles of equal area. Traffic , when reaching an intersection, may only go straight ahead, or turn left or right through $120^0$ , as shown in the diagram. [img]https://cdn.artofproblemsolving.com/attachments/3/6/a100a5c39bf15116582bc0bceb76fcbae28af9.png[/img] No turns are permitted except the ones at intersections . One car departs for a certain nearby intersection and when it reaches it a second car starts moving toward it. From then on both cars continue travelling at the same speed (but do not necessarily take the same turns). Is it possible that there will be a time when they will encounter each other somewhere? ( N . N . Konstantinov , Moscow )

For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$. Find $I_1+I_2+I_3+I_4$. [i]1992 University of Fukui entrance exam/Medicine[/i]

Prove that for all n=3,4,5.... there excist odd x,y such $2^n=x^2 + 7y^2$ .

Find all positive integers $n$ for which the number $\frac{n^{3n-2}-3n+1}{3n-2}$ is whole. [hide=EDIT:] In the original problem instead of whole we search for integers, so with this change $n=1$ will be a solution. [/hide]

The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.