Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$

For \(u,v \in\mathbb{R}^4\), let \(<u,v>\) denote the usual dot product. Define a [i]vector field[/i] to be a map \(\omega:\mathbb{R}\to\mathbb{R}\) such that \(<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.\) Find a maximal collection of vector fields \(\left\{\omega_1,...,\omega_k\right\}\) such that the map \(\Omega\) sending \(z\) to \(\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z)\), with \(\lambda_1,\ldots,\lambda_k\in\mathbb{R}\), is nonzero on \(\mathbb{R}^4\backslash\{0\}\) unless \(\lambda_1=\cdots=\lambda_k=0\)

The $5$-digit number $\underline{2}$ $\underline{0}$ $\underline{1}$ $\underline{8}$ $\underline{U}$ is divisible by $9$. What is the remainder when this number is divided by $8$? $\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?

Prove that there is no natural number $k$ such that $k^{1999} - k^{1998} = 2k + 2$.

What is the smallest positive integer with exactly $7$ distinct proper divisors?

In the eight-term sequence $A, B, C, D, E, F, G, H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A + H$? $ \textbf{(A)}\ 17 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 26 \qquad \textbf{(E)}\ 43 $

[b]9.[/b] Let $f(z)= z^n +a_1 z^{n-1}+\dots + a_n$ be a polynomial over the field of the complex numbers and let $E_f$ denote the closed (not necessarily connected) domain of complex numbers $z$ for which $\mid f(z) \mid \leq 1$. Show that there exists a point $z_0 \in E_f$ such that $\mid f'(z_0) \mid \geq n$. [b](F. 5)[/b]

Let $f$ be a function defined from $((x,y) : x,y$ real, $xy\ne 0)$ to the set of all positive real numbers such that $ (i) f(xy,z)= f(x,z)\cdot f(y,z)$ for all $x,y \ne 0$ $ (ii) f(x,yz)= f(x,y)\cdot f(x,z)$ for all $x,y \ne 0$ $ (iii) f(x,1-x) = 1 $ for all $x \ne 0,1$ Prove that $ (a) f(x,x) = f(x,-x) = 1$ for all $x \ne 0$ $(b) f(x,y)\cdot f(y,x) = 1 $ for all $x,y \ne 0$ The condition (ii) was left out in the paper leading to an incomplete problem during contest.

There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does [i]not[/i] necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

Show that the sum of the first $n$ terms in the binomial expansion of $(2-1)^{-n}$ is $\frac{1}{2},$ where $n$ is a positive integer.

Let $n\in \mathbb{N}$ be a number multiple of 4. We take all permutations $(a_1,a_2...a_n)$ of the numbers $(1,2...n)$, for which $\forall j$, $a_i+j=n+1$ where $i=a_j$. Prove that there exist $\frac{(\frac{1}{2}n)!}{(\frac{1}{4}n)!}$ such permutations.

Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

A positive integer $N$ is \emph{piquant} if there exists a positive integer $m$ such that if $n_i$ denotes the number of digits in $m^i$ (in base $10$), then $n_1+n_2+\cdots + n_{10}=N$. Let $p_M$ denote the fraction of the first $M$ positive integers that are piquant. Find $\lim\limits_{M\to \infty} p_M$. [i]Proposed by James Lin.[/i]

Triangle $ABC$ is given, where $AC<BC$ and $\angle ACB=60^\circ\!\!.$ Point $D$, distinct from $A$, lies on the segment $AC$ such that $AB=BD$, and point $E$, distinct from $B$, lies on the line $BC$ such that $AB=AE$. Prove that $\angle DEC=30^\circ$.

A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.

Given is a half circle with midpoint $O$ and diameter $AB$. Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle. If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$.

We call polynomials $A(x) = a_n x^n +. . .+a_1 x+a_0$ and $B(x) = b_m x^m +. . .+b_1 x+b_0$ ($a_n b_m \neq 0$) similar if the following conditions hold: $(i)$ $n = m$; $(ii)$ There is a permutation $\pi$ of the set $\{ 0, 1, . . . , n\} $ such that $b_i = a_{\pi (i)}$ for each $i \in {0, 1, . . . , n}$. Let $P(x)$ and $Q(x)$ be similar polynomials with integer coefficients. Given that $P(16) = 3^{2012}$, find the smallest possible value of $|Q(3^{2012})|$. [i]Proposed by Milos Milosavljevic[/i]

Let $a_1,a_2,\ldots,a_{18}$ be a list of prime numbers such that $\frac{1}{64}\times a_1\times a_2\times\cdots\times a_{18}$ is one million. Determine the sum of all positive integers $n$ such that $$\sum_{i=1}^{18}\frac{1}{\log_{a_i}n}$$ is a positive integer. [i]2017 CCA Math Bonanza Individual Round #8[/i]

The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $ 1$ to $ 15$ in clockwise order. Committee rules state that a Martian must occupy chair $ 1$ and an Earthling must occupy chair $ 15$. Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is $ N\cdot (5!)^3$. Find $ N$.

Let $x,y,z$ be positive numbers such that $x^2+y^2+z^2=1$. Prove that $$\frac x{1-x^2}+\frac y{1-y^2}+\frac z{1-z^2}\ge\frac{3\sqrt3}2$$

Given a triangle $ABC$, $A',B',C'$ are the midpoints of $\overline{BC},\overline{AC},\overline{AB}$, respectively. $B^*,C^*$ lie in $\overline{AC},\overline{AB}$, respectively, such that $\overline{BB^*},\overline{CC^*}$ are the altitudes of the triangle $ABC$. Let $B^{\#},C^{\#}$ be the midpoints of $\overline{BB^*},\overline{CC^*}$, respectively. $\overline{B'B^{\#}}$ and $\overline{C'C^{\#}}$ meet at $K$, and $\overline{AK}$ and $\overline{BC}$ meet at $L$. Prove that $\angle{BAL}=\angle{CAA'}$

Given several numbers, one of them, $a$, is chosen and replaced by the three numbers $\frac{a}{3}, \frac{a}{3}, \frac{a}{3}$. This process is repeated with the new set of numbers, and so on. Originally, there are $1000$ ones, and we apply the process several times. A number $m$ is called [i]good[/i] if there are $m$ or more numbers that are the same after each iteration, no matter how many or what operations are performed. Find the largest possible good number.

How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?

We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF,\, DE$ intersect $AB,\, AC$ at points $M,\, N$, respectively. Denote $(I_1),\, (I_2)$ by the circumcircles of triangles $DEM,\, DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1),\, (I_2)$ different from $D$. $Q$ is the intersection of $(J_1),\, (J_2)$ different from $D$. a. Prove that all points $D,\, P,\, Q$ lie on the same line. b. The circumcircles of triangles $AEF,\, AHK$ intersect at $A,\, G$. $(AEF)$ also cut $AQ$ at $A,\, L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$.