Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$

Four circles of radius $1$ are each tangent to two sides (line segments) of a square and externally tangent to a circle of radius $3$. What is the area of the space that is inside the square but not contained in any of the circles?

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$ Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$

in $ABC$ let $E$ and $F$ be points on line $AC$ and $AB$ respectively such that $BE$ is parallel to $CF$. suppose that the circumcircle of $BCE$ meet $AB$ again at $F'$ and the circumcircle of $BCF$ meets $AC$ again at $E'$. show that $BE'$ Is parallel to $CF'$.

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\] for all $x,y\in\mathbb{R}$.

If the operation $x * y$ is defined by $x * y = (x+1)(y+1) - 1$, then which one of the following is FALSE? $\textbf{(A)} \ x * y = y *x$ for all real $x$ and $y$. $\textbf{(B)} \ x * (y + z) = ( x * y ) + (x * z)$ for all real $x,y,$ and $z$ $\textbf{(C)} \ (x-1) * (x+1) = (x * x) - 1$ for all real $x$. $\textbf{(D)} \ x * 0 = x$ for all real $x$. $\textbf{(E)} \ x * (y * z) = (x * y) * z$for all real $x,y,$ and $z$.

Let $a_1,a_2,\dots, a_{17}$ be a permutation of $1,2,\dots, 17$ such that $(a_1-a_2)(a_2-a_3)\dots(a_{17}-a_1)=n^{17}$ .Find the maximum possible value of $n$ .

Two triangles are drawn on a plane in such a way that the area covered by their union is an n-gon (not necessarily convex). Find all possible values of the number of vertices n.

On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$

An archery target can be represented as three concentric circles with radii $3$, $2$, and $1$ which split the target into $3$ regions, as shown in the figure below. What is the area of Region $1$ plus the area of Region $3$? [i]2015 CCA Math Bonanza Team Round #1[/i]

How many pairs of positive integers $ (a,b)$ are there such that $ \gcd(a,b) \equal{} 1$ and \[ \frac {a}{b} \plus{} \frac {14b}{9a} \]is an integer? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ \text{infinitely many}$

The base three representation of $x$ is \[ 12112211122211112222. \]The first digit (on the left) of the base nine representation of $x$ is $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 5$

If $p_k>0,a_k\ge2~(k=1,2,\ldots,n)$ and $$S_n=\sum_{k=1}^na_k,A_n=\prod_{\text{cyc}}a_1^{p_2+p_3+\ldots+p^n},B_n=\prod_{k=1}^na_k^{p_k},$$ then prove that $$\sum_{k=1}^np_k\log_{S_n-a_k}a_k\ge\left(\sum_{k=1}^np_k\right)\log_{A_n}B_n$$ [i]Proposed by Mihály Bencze[/i]

Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and \[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\] $x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.

There are two pipes on the plane (the pipes are circular cylinders of equal size, $4$ m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of the surface of this chamber.

In a convex quadrilateral $ABCD$ suppose $AC \cap BD = O$ and $M$ is the midpoint of $BC$. Let $MO \cap AD = E$. Prove that $\frac{AE}{ED} = \frac{S_{\triangle ABO}}{S_{\triangle CDO}}$.

Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that \[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\] Find $a^3+b^3+c^3+d^3$. [i]Proposed by Connor Gordon[/i]

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.

Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.

Consider all rational numbers of the form $\tfrac{p}{q}$ where $p,q$ are relatively prime positive integers less than or equal to $8$, and plot them on the $xy$-plane, where $\tfrac{p}{q}$ corresponds to point $(p,q)$. Arrange the rationals in increasing order $\{P_1,P_2,\dots,P_n\}$ and form a polygon by connecting points $P_i$ and $P_{i+1}$ for $1\le i<n$ and connecting both $P_1$ and $P_n$ to the origin. What is the area of the polygon?

Count the sum of the four-digit positive integers containing only odd digits in their decimal representation.

Show that the only integral solution to \[\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1\\ x + y + z = 3n \\ \end{array} \right. \] with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.

Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane. (Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version [url=https://artofproblemsolving.com/community/c6h2094701p15140671]here[/url]) [hide=note]There was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.[/hide]

Let $f\in C^1(a,b)$, $\lim_{x\to a^+}f(x)=\infty$, $\lim_{x\to b^-}f(x)=-\infty$ and $f'(x)+f^2(x)\geq -1$ for $x\in (a,b)$. Prove that $b-a\geq\pi$ and give an example where $b-a=\pi$.