Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$

The real numbers $x$ and $y$ are such that for any distinct odd primes $p$ and $q$ the number $x^p + y^q$ is rational. Prove that $x$ and $y$ are rational numbers.

Circle $k_1$ has radius $10$, externally touching circle $k_2$ with radius $18$. Circle $k_3$ touches both circles, as well as the line $e$ determined by their centres. Let $k_4$ be the circle touching $k_2$ and $k_3$ externally (other than $k_1$) whose center lies on line $e$. What is the radius of $k_4$?

Let $ABCD$ be a rectangle with $AB=8$ and $AD=20$. Two circles of radius $5$ are drawn with centers in the interior of the rectangle - one tangent to $AB$ and $AD$, and the other passing through both $C$ and $D$. What is the area inside the rectangle and outside of both circles?

Let an acute-angled triangle $ABC$'s circumcircle is $S$. $S$'s tangent from $B$ and $C$ intersects at point $M$. A line, lies $M$ and parallel to $[AB]$ intersects with $S$ at points $D$ and $E$, intersect with $[AC]$ at point $F$. Prove that $$[DF]=[FE]$$

$E=\{1,2,\cdots,200\},G=\{a_1,a_2,\cdots,a_{100}\}\subset E$. $G$ satisfies the following: (1)For any $1\geq i<j\geq100$, a_i+a_j\neq201. (2)$\sum_{i=1}^{100}a_i=10080$. Prove that the number of odd numbers in $G$ is a multiple of $4$, and the sum the square of all numbers in $G$ is fixed.

Prove that infinitely many positive integers cannot be written in the form \[{x_{1}}^{3}+{x_{2}}^{5}+{x_{3}}^{7}+{x_{4}}^{9}+{x_{5}}^{11},\] where $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\in \mathbb{N}$.

Harry Potter would like to purchase a new owl which cost him 2 Galleons, a Sickle, and 5 Knuts. There are 23 Knuts in a Sickle and 17 Sickles in a Galleon. He currently has no money, but has many potions, each of which are worth 9 Knuts. How many potions does he have to exhange to buy this new owl? [i]2015 CCA Math Bonanza Individual Round #7[/i]

The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased? [i]Ray Li[/i]

Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.

Determine all integers $n\ge1$ for which there exists $n$ real numbers $x_1,\ldots,x_n$ in the closed interval $[-4,2]$ such that the following three conditions are fulfilled: - the sum of these real numbers is at least $n$. - the sum of their squares is at most $4n$. - the sum of their fourth powers is at least $34n$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?

Each of 2010 boxes in a line contains a single red marble, and for $ 1 \le k \le 2010$, the box in the $ kth$ position also contains $ k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $ P(n)$ be the probability that Isabella stops after drawing exactly $ n$ marbles. What is the smallest value of $ n$ for which $ P(n) < \frac {1}{2010}$? $ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

$\frac{1-\frac{1}{3}}{1-\frac{1}{2}} =$ $\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{3}{2} \qquad \text{(E)}\ \frac{4}{3}$

Aprameya graphs the equation $2x = y + 4$ on the coordinate plane. It turns out that there is a unique point with a positive integer coordinate and a negative integer coordinate lying on Aprameya's graph. What is the sum of the coordinates of this point? $\textbf{(A) }{-}3\qquad\textbf{(B) }{-}1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }2$

Let $ABC$ be a triangle with side lengths $AB=6, AC=7,$ and $BC=8.$ Let $H$ be the orthocenter of $\triangle ABC$ and $H'$ be the reflection of $H$ across the midpoint $M$ of $BC.$ $\tfrac{[ABH']}{[ACH']}$ can be expressed as $\frac{p}{q}$. Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #14[/i]

Characterize those configurations of $ n$ coplanar straight lines for which the sum of angles between all pairs of lines is maximum. [i]L.Fejes-Toth, A. Heppes[/i]

$n$ points are given on a circle $\omega$. There is a circle with radius smaller than $\omega$ such that all these points lie inside or on the boundary of this circle. Prove that we can draw a diameter of $\omega$ with endpoints not belonging to the given points such that all the $n$ given points remain in one side of the diameter.

Prove the following inequality. \[ \sum_{n=0}^\infty \int_0^1 x^{4011} (1-x^{2006})^\frac{n-1}{2006}\ dx<\frac{2006}{2005} \]

Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2=c^2$, solve the system: \[ z^2=x^2+y^2 \] \[ (z+c)^2=(x+a)^2+(y+b)^2 \] in real numbers $x, y$ and $z$.

Problems 17, 18, and 19 refer to the following: At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.) $ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 5\qquad\text{(D)}\ 7\qquad\text{(E)}\ 15 $

Let $D$ be the set of positive reals different from $1$ and let $n$ be a positive integer. If for $f: D\rightarrow \mathbb{R}$ we have $x^n f(x)=f(x^2)$, and if $f(x)=x^n$ for $0<x<\frac{1}{1989}$ and for $x>1989$, then prove that $f(x)=x^n$ for all $x \in D$.

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Prove that, among $100000$ consecutive $100$-digit positive integers, there is an integer $n$ such that the length of the period of the decimal expansion of $\frac1n$ is greater than $2011$.

Let $r$ be a nonzero real number. The values of $z$ which satisfy the equation \[ r^4z^4 + (10r^6-2r^2)z^2-16r^5z+(9r^8+10r^4+1) = 0 \] are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of $r$, and find this area.