Prove that if $1 < a < b < c$, then $$\log_a(\log_a b) + \log_b(\log_b c) + \log_c(\log_c a) > 0.$$

For each positive integer $n,$ let $f(n)$ be either the unique integer $r \in \{0,1, \ldots, n-1\}$ such that $n$ divides $15r-1,$ or $0$ if such $r$ does not exist. Compute $$f(16)+f(17)+f(18)+\cdots+f(300).$$

Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

In a regular hexagon $ABCDEF$ of side length $8$ and center $K$, points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$. Find the area of pentagon $WBCUK$. [i]Proposed by Bradley Guo[/i]

Prove that a bounded figure cannot have more than one center of symmetry.

Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.

CCA's B building has 6 rooms on the second floor, labeled B201 to B206, as well as 8 rooms on the first floor, labeled B101 to B108. Annie is currently in room B205. Each minute, she chooses to stay or change floors with equal probability, and chooses a classroom on that floor to go to at random (she can stay in the classroom that she's already in). B104, B108, and B203 are the only rooms that have teachers who will scold her for randomly walking around during class time. The probability that she is first scolded in room B203 can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #2[/i]

The squares in a $9\times 9$ grid are numbered from $11$ to $99$, where the first digit is the row and the second the column. Each square is colored black or white. Squares $44$ and $49$ are black. Every black square shares an edge with at most one other black square, and each white square shares an edge with at most one other white square. What color is square $99$?

Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$? $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$

Find all pairs of positive integers $(x,y)$ so that $$\frac{(x^2-x+1)(y^2-y+1)}{xy}\in\mathbb N.$$

Let $S_0 = 0, S_1 = 1,$ and for $n \ge 2,$ let $S_n = S_{n-1}+5S_{n-2}.$ What is the sum of the five smallest primes $p$ such that $p \mid S_{p-1}$?

A positive integer $N$ is divided in $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$ The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.

Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$.

Let $ABC$ be a triangle with $\angle B - \angle C = 30^{\circ}$. Let $D$ be the point where the $A$-excircle touches line $BC$, $O$ the circumcenter of triangle $ABC$, and $X,Y$ the intersections of the altitude from $A$ with the incircle with $X$ in between $A$ and $Y$. Suppose points $A$, $O$ and $D$ are collinear. If the ratio $\frac{AO}{AX}$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $\gcd(a,b,d)=1$ and $c$ not divisible by the square of any prime, find $a+b+c+d$. [i]James Tao[/i]

Let $ n \geq 2$ be an integer. Find the largest real number $ \lambda$ such that the inequality \[ a^2_n \geq \lambda \sum^{n\minus{}1}_{i\equal{}1} a_i \plus{} 2 \cdot a_n.\] holds for any positive integers $ a_1, a_2, \ldots a_n$ satisfying $ a_1 < a_2 < \ldots < a_n.$

The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$. [img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]

Let $ABC$ be a triangle and $\Omega$ its circumcircle. Let the internal angle bisectors of $\angle BAC, \angle ABC, \angle BCA$ intersect $BC,CA,AB$ on $D,E,F$, respectively. The perpedincular line to $EF$ through $D$ intersects $EF$ on $X$ and $AD$ intersects $EF$ on $Z$. The circle internally tangent to $\Omega$ and tangent to $AB,AC$ touches $\Omega$ on $Y$. Prove that $(XYZ)$ is tangent to $\Omega$.

For positive integer $a,b,c,d$ there are $a+b+c+d$ points on plane which none of three are collinear. Prove there exist two lines $l_1, l_2 $ such that (1) $l_1, l_2 $ are not parallel. (2) $l_1, l_2 $ do not pass through any of $a+b+c+d$ points. (3) There are $ a, b, c, d $ points on each region separated by two lines $l_1, l_2 $.

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

Alice is stacking balls on the ground in three layers using two sizes of balls: small and large. All small balls are the same size, as are all large balls. For the first layer, she uses $6$ identical large balls $A, B, C, D, E$, and $F$ all touching the ground and so that $D, E, F$ touch each other, A touches $E$ and $F$, $B$ touches $D$ and $F$, and $C$ touches $D$ and $E$. For the second layer, she uses $3$ identical small balls, $G, H$, and $I$; $G$ touches $A, E$, and $F, H$ touches $B, D$, and $F$, and $I$ touches $C, D$, and $E$. Obviously, the small balls do not intersect the ground. Finally, for the top layer, she uses one large ball that touches $D, E, F, G, H$, and $I$. If the large balls have volume $2015$, the sum of the volumes of all the balls in the pyramid can be written in the form $a\sqrt{b}+c$ for integers $a, b, c$ where no integer square larger than $1$ divides $b$. What is $a + b + c$? (This diagram may not have the correct scaling, but just serves to clarify the layout of the problem.) [asy] size(6cm); pair A, B, C, D, E, F; A = (0,0); F = (1,0); B = (2,0); E = rotate(60, A)*F; D = F + E; C = rotate(60, A)*B; draw(Circle(A, 0.5), mediumblue); draw(Circle(B, 0.5), mediumblue); draw(Circle(C, 0.5), mediumblue); draw(Circle(D, 0.5), mediumblue); draw(Circle(E, 0.5), mediumblue); draw(Circle(F, 0.5), mediumblue); pair G = (E+A+F)/3; pair I = (C+E+D)/3; pair H = (D+B+F)/3; draw(Circle(G, 0.25), mediumblue); draw(Circle(I, 0.25), mediumblue); draw(Circle(H, 0.25), mediumblue); label("A", A, fontsize(10pt)); label("B", B, fontsize(10pt)); label("C", C, fontsize(10pt)); label("D", D, fontsize(10pt)); label("E", E, fontsize(10pt)); label("F", F, fontsize(10pt)); label("G", G, fontsize(5pt)); label("I", I, fontsize(5pt)); label("H", H, fontsize(5pt)); label("Figure 1: The projection of the balls onto the ground", (1,-1), fontsize(10pt)); [/asy]

The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.

There is a set of $2n$ chips of $n$ different colors, two chips of each color. The chips are randomly placed in a row. Prove that the probability that there are two adjacent chips of the same color in a row is greater than $1/2$. [i]From the folklore[/i]

Find all pairs $(n, p)$ that satisfy the following condition, where $n$ is a positive integer and $p$ is a prime number. [b]Condition)[/b] $2n-1$ is a divisor of $p-1$ and $p$ is a divisor of $4n^2+7$.

[b]9.[/b] Let $A_1, \dots , A_n$ and $B$ be ideals of an assoticative ring $R$ such that $B$ is contained in the set-union of the ideals $A_i$($i=1, \dots , n$) but not contained in the union of any $n-1$ of the ideals $A_i$. Show that, for some positive integer $k$, $B_k$ is contained in the intersection of the ideals $A_i$. [b](A. 19)[/b]