Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip? $\textbf{(A)} \text{ 60} \qquad \textbf{(B)} \text{ 72} \qquad \textbf{(C)} \text{ 92} \qquad \textbf{(D)} \text{ 96} \qquad \textbf{(E)} \text{ 120}$

Let $f(x)$ is an odd function on $R$ , $f(1)=1$ and $f(\frac{x}{x-1})=xf(x)$ $(\forall x<0)$. Find the value of $f(1)f(\frac{1}{100})+f(\frac{1}{2})f(\frac{1}{99})+f(\frac{1}{3})f(\frac{1}{98})+\cdots +f(\frac{1}{50})f(\frac{1}{51}).$

The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$ and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.

Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer. [i]Laurentiu Panaitopol, Romania[/i]

There are 105 users on the social media platform Mathsenger, every pair of which has a direct messaging channel. Prove that each messaging channel may be assigned one of 100 encryption keys, such that no 4 users have the 6 pairwise channels between them all being assigned the same encryption key. [i]Proposed by Fredy Yip[/i]

A square with side $1$ contains a non-self-intersecting polyline of length at least $200$. Prove that there is a straight line parallel to the side of the square that has at least $101$ points in common with this polyline.

Let the sequence $a_n$, $n\geq2$, $a_n=\frac{\sqrt[3]{n^3+n^2-n-1}}{n} $. Find the greatest natural number $k$ ,such that $a_2 \cdot a_3 \cdot . . .\cdot a_k <8$

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Is it possible to write the numbers $17$,$18$,$19$,...$32$ in a $4*4$ grid of unit squares with one number in each square such that if the grid is divided into four $2*2$ subgrids of unit squares ,then the product of numbers in each of the subgrids divisible by $16$?

Angles $\alpha, \beta, \gamma$ satisfy the inequality $\sin \alpha +\sin \beta +\sin \gamma \ge 2$. Prove that $\cos \alpha + \cos \beta +\cos \gamma \le \sqrt5.$

Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.

Let $$A = \{2, 4, \ldots, 1000\},$$ $$B = \{3, 6, \ldots, 999\},$$ $$C = \{5, 10, \ldots, 1000\},$$ $$D = \{7, 14, \ldots, 994\},$$ $$E = \{11, 22, \ldots, 990\},$$ $$\textrm{and } F = \{13, 26, \ldots, 988\}.$$ Find the number of elements in the set $(((((A\cup B)\cap C)\cup D)\cap E)\cup F)$. [i]2022 CCA Math Bonanza Individual Round #7[/i]

Evaluate $$\lim_{n\to \infty} \frac{1}{n^4 } \prod_{i=1}^{2n} (n^2 +i^2 )^{\frac{1}{n}}.$$

$f$ is a function whose domain is the set of nonnegative integers and whose range is contained in the set of nonnegative integers. $f$ satisfies the condition that $f(f(n))+f(n)=2n+3$ for all nonnegative integers $n$. Find $f(2014)$.

Let $ABC$ be an isosceles triangle with $AB=4$, $BC=CA=6$. On the segment $AB$ consecutively lie points $X_{1},X_{2},X_{3},\ldots$ such that the lengths of the segments $AX_{1},X_{1}X_{2},X_{2}X_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{4}$. On the segment $CB$ consecutively lie points $Y_{1},Y_{2},Y_{3},\ldots$ such that the lengths of the segments $CY_{1},Y_{1}Y_{2},Y_{2}Y_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. On the segment $AC$ consecutively lie points $Z_{1},Z_{2},Z_{3},\ldots$ such that the lengths of the segments $AZ_{1},Z_{1}Z_{2},Z_{2}Z_{3},\ldots$ form an infinite geometric progression with starting value $3$ and common ratio $\frac{1}{2}$. Find all triplets of positive integers $(a,b,c)$ such that the segments $AY_{a}$, $BZ_{b}$ and $CX_{c}$ are concurrent.

A quadruplet of distinct positive integers $(a, b, c, d)$ is called $k$-good if the following conditions hold: 1. Among $a, b, c, d$, no three form an arithmetic progression. 2. Among $a+b, a+c, a+d, b+c, b+d, c+d$, there are $k$ of them, forming an arithmetic progression. $a)$ Find a $4$-good quadruplet. $b)$ What is the maximal $k$, such that there is a $k$-good quadruplet?

Given an acute angle $APX$ in the plane, construct a square $ABCD$ such that $P$ lies on the side $BC$ and ray $PX$ meets $CD$ in a point $Q$ such that $AP$ bisects the angle $BAQ$.

Show that \[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\] for all positive real numbers $a$ and $b$ such that $ab\geq 1$.

A car travels $120$ miles from $A$ to $B$ at $30$ miles per hour but returns the same distance at $40$ miles per hour. The average speed for the round trip is closest to: $\textbf{(A)}\ 33\text{ mph} \qquad \textbf{(B)}\ 34\text{ mph} \qquad \textbf{(C)}\ 35\text{ mph} \qquad \textbf{(D)}\ 36\text{ mph} \qquad \textbf{(E)}\ 37\text{ mph}$

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$. Anna and Bob play a game on this grid. Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first. 1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line. 2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted. As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins. Does Anna have a winning strategy? [i]Proposed by Demetres Christofides, Cyprus[/i]

A company of $n$ soldiers is such that (i) $n$ is a palindrome number (read equally in both directions); (ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively. Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.

Points $ A$ and $ B$ lie on a circle centered at $ O$, and $ \angle AOB\equal{}60^\circ$. A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle? $ \textbf{(A)}\ \frac{1}{16} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{4}$