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 $b-a=\pi$.

A natural number $n$ is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it $a_0$, $a_1$, $a_2$, $...$, $a_{2n-1}$ the polynomial $x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0$ has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)

Let $x$, $y$, $z > 0$ such that $x^2 + y^2 + z^2 = 3$. Then $$x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}$$

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

A square \( K = 2025 \times 2025 \) is given. We define a [i]stick[/i] as a rectangle where one of its sides is \( 1 \), and the other side is a positive integer from \( 1 \) to \( 2025 \). Find the largest positive integer \( C \) such that the following condition holds: [list] [*] If several sticks with a total area not exceeding \( C \) are taken, it is always possible to place them inside the square \( K \) so that each stick fully completely covers an integer number of \( 1 \times 1 \) squares, and no \( 1 \times 1 \) square is covered by more than one stick. [/list] [i](Basically, you can rotate sticks, but they have to be aligned by lines of the grid)[/i] [i]Proposed by Anton Trygub[/i]

A table consisting of $9$ rows and $2001$ columns is filfed with integers $1,2,..., 2001$ in such a way that each of these integers occurs in the table exactly $9$ times and the integers in any column differ by no more than $3$. Find the maximum possible value of the minimal column sum (sum of the numbers in one column).

Assume that $a$ is a given irrational number. (a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$. (b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.

At a gala banquet, $12n + 6$ chairs, where $n \in N$, are equally arranged around a large round table. A seating will be called a proper seating of rank $n$ if a gathering of $6n + 3$ married couples sit around this table such that each seated person also has exactly one sibling (brother/sister) of the opposite gender present (siblings cannot be married to each other) and each man is seated closer to his wife than his sister. Among all proper seats of rank n find the maximum possible number of women seated closer to their brother than their husband. (The maximum is taken not only across all possible seating arrangements for a given gathering, but also across all possible gatherings.)

Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called [i]simple[/i] if it satisfies $(x \times y) \times z = x \times (y \times z)$ for all $x,y,z \in S$ and $x \times y \in \{x,y\}$ for all $x,y \in S$. Given a simple operation $\times$ on $S$, any string of elements in $S$ can be reduced to a single element, such as $xyz \to x \times (y \times z)$. A string of variables in $S$ is called[i] full [/i]if it contains each variable in $S$ at least once, and two strings are [i]equivalent[/i] if they evaluate to the same variable regardless of which simple $\times$ is chosen. For example $xxx$, $xx$, and $x$ are equivalent, but these are only full if $n=1$. Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$. Determine the number of elements of $T$.

Given two positive numbers $a,b$ such that $a^3 +b^3 = a^5 +b^5$, then the greatest value of $M = a^2 + b^2 - ab$ is (A): $\frac14$ (B): $\frac12$ (C): $2$ (D): $1$ (E): None of the above.

Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.