Eric rolls a ten-sided die (with sides labeled $1$ through $10$) repeatedly until it lands on $3, 5$, or $7$. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Suppose $z=a+bi$ is a solution of the polynomial equation $c_4z^4+ic_3z^3+c_2z^2+ic_1z+c_0=0$, where $c_0$, $c_1$, $c_2$, $c_3$, $a$, and $b$ are real constants and $i^2=-1$. Which of the following must also be a solution? $\textbf{(A) } -a-bi\qquad \textbf{(B) } a-bi\qquad \textbf{(C) } -a+bi\qquad \textbf{(D) }b+ai \qquad \textbf{(E) } \text{none of these}$

Ivan and Kolya play a game, Ivan starts. Initially, the polynomial $x-1$ is written of the blackboard. On one move, the player deletes the current polynomial $f(x)$ and replaces it with $ax^{n+1}-f(-x)-2$, where $\deg(f)=n$ and $a$ is a real root of $f$. The player who writes a polynomial which does not have real roots loses. Can Ivan beat Kolya?

What is the product of all the solutions to the equation $$\log_{7x}2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?$$ $\textbf{(A) }(\log_{2023}7 \cdot \log_{2023}289)^2 \qquad\textbf{(B) }\log_{2023}7 \cdot \log_{2023}289\qquad\textbf{(C) }1\qquad\textbf{(D) }\log_{7}2023 \cdot \log_{289}2023\qquad\textbf{(E) }(\log_{7}2023 \cdot \log_{289}2023)^2$

Each spinner is divided into $3$ equal parts. The results obtained from spinning the two spinners are multiplied. What is the probability that this product is an even number? [asy] draw(circle((0,0),2)); draw(circle((5,0),2)); draw((0,0)--(sqrt(3),1)); draw((0,0)--(-sqrt(3),1)); draw((0,0)--(0,-2)); draw((5,0)--(5+sqrt(3),1)); draw((5,0)--(5-sqrt(3),1)); draw((5,0)--(5,-2)); fill((0,5/3)--(2/3,7/3)--(1/3,7/3)--(1/3,3)--(-1/3,3)--(-1/3,7/3)--(-2/3,7/3)--cycle,black); fill((5,5/3)--(17/3,7/3)--(16/3,7/3)--(16/3,3)--(14/3,3)--(14/3,7/3)--(13/3,7/3)--cycle,black); label("$1$",(0,1/2),N); label("$2$",(sqrt(3)/4,-1/4),ESE); label("$3$",(-sqrt(3)/4,-1/4),WSW); label("$4$",(5,1/2),N); label("$5$",(5+sqrt(3)/4,-1/4),ESE); label("$6$",(5-sqrt(3)/4,-1/4),WSW); [/asy] $\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ \frac{7}{9} \qquad \text{(E)}\ 1$

For a real-valued function $f(x,y)$ of two positive real variables $x$ and $y$, define $f$ to be [i]linearly bounded[/i] if and only if there exists a positive number $K$ such that $|f(x,y)| < K(x+y)$ for all positive $x$ and $y.$ Find necessary and sufficient conditions on the real numbers $\alpha$ and $\beta$ such that $x^{\alpha}y^{\beta}$ is linearly bounded.

Let $X_1,X_2,\ldots$ be a sequence of independent random variables distributed exponentially with mean $1$. Suppose $\mathbb N$ is a random variable independent of $X_i$'s that has a Poisson distribution with mean $\lambda>0$. What is the expected value of $X_1+X_2+\ldots+X_N$? $\textbf{(A)}~N^2$ $\textbf{(B)}~\lambda+\lambda^2$ $\textbf{(C)}~\lambda^2$ $\textbf{(D)}~\lambda$

Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that 1) $f(0,x)$ is non-decreasing ; 2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ; 3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ; 4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .

Using white cubes of side $1$, a prism (without holes) was assembled. The faces of the prism were painted black. It is known that the cubes left with exactly $4$ white faces are $20$ in total. Determine what the dimensions of the prism can be. Give all the possibilities.

In acute triangle $ABC$, $\angle{A}<\angle{B}$ and $\angle{A}<\angle{C}$. Let $P$ be a variable point on side $BC$. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively, such that $BP=PD$ and $CP=PE$. Prove that as $P$ moves along side $BC$, the circumcircle of triangle $ADE$ passes through a fixed point other than $A$.

There are ten apples and $p$ pears in a basket. Anna eats two apples, and she finds that there are now more pears than apples. She then eats four pears. After eating the pears, she notices that there are more apples than pears. What is the sum of all possible values of $p$? $\textbf{(A) } 19\qquad\textbf{(B) } 28\qquad\textbf{(C) } 30\qquad\textbf{(D) } 42\qquad\textbf{(E) } 45$

Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?

In five years, Tom will be twice as old as Cindy. Thirteen years ago, Tom was three times as old as Cindy. How many years ago was Tom four times as old as Cindy?

Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$

What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $ Where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set {$ 1,2,3,4 $}?

Let $A$ be a fixed point of a circle $\omega$. Let $BC$ be an arbitrary chord of $\omega$ passing through a fixed point $P$. Prove that the nine-points circles of triangles $ABC$ touch some fixed circle not depending on $BC$.