If $3 = k \cdot 2^r$ and $15 = k \cdot 4^r$, then $r =$ $\text{(A)}\ - \log_2 5 \qquad \text{(B)}\ \log_5 2 \qquad \text{(C)}\ \log_{10} 5 \qquad \text{(D)}\ \log_2 5 \qquad \text{(E)}\ \displaystyle \frac{5}{2}$

In line for a SuperRockPop concert were 2005 people. With the aim of offering $3$ tickets for the "backstage", the first person in line was asked to shout "Super", ` the second "Rock", ` the third "Pop", ` the fourth "Super", ` the fifth "Rock", ` the sixth "Pop" and so on. Anyone who said "Rock" or "Pop" was eliminated. This process was repeated, always starting from the first person in the new line, until only $3$ people remained. What positions were these people in at the beginning?

Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.

Solve the equation $ \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2$.

A deck of $ n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $ (n\plus{}1)/2$.

If $a$ and $b$ are positive numbers, prove the inequality $$a^2 +ab+b^2\ge 3(a+b-1).$$

Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that $$P(x^{n},x^{n+1},x+x^{n+2})=x.$$

The sides of the triangle $ABC$ satisfy the relation $c^2 = a^2 + b^2$. Show that angle $C$ is right.

Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.

Jeff rotates spinners $ P$, $ Q$ and $ R$ and adds the resulting numbers. What is the probability that his sum is an odd number? [asy]size(200); path circle=circle((0,0),2); path r=(0,0)--(0,2); draw(circle,linewidth(1)); draw(shift(5,0)*circle,linewidth(1)); draw(shift(10,0)*circle,linewidth(1)); draw(r,linewidth(1)); draw(rotate(120)*r,linewidth(1)); draw(rotate(240)*r,linewidth(1)); draw(shift(5,0)*r,linewidth(1)); draw(shift(5,0)*rotate(90)*r,linewidth(1)); draw(shift(5,0)*rotate(180)*r,linewidth(1)); draw(shift(5,0)*rotate(270)*r,linewidth(1)); draw(shift(10,0)*r,linewidth(1)); draw(shift(10,0)*rotate(60)*r,linewidth(1)); draw(shift(10,0)*rotate(120)*r,linewidth(1)); draw(shift(10,0)*rotate(180)*r,linewidth(1)); draw(shift(10,0)*rotate(240)*r,linewidth(1)); draw(shift(10,0)*rotate(300)*r,linewidth(1)); label("$P$", (-2,2)); label("$Q$", shift(5,0)*(-2,2)); label("$R$", shift(10,0)*(-2,2)); label("$1$", (-1,sqrt(2)/2)); label("$2$", (1,sqrt(2)/2)); label("$3$", (0,-1)); label("$2$", shift(5,0)*(-sqrt(2)/2,sqrt(2)/2)); label("$4$", shift(5,0)*(sqrt(2)/2,sqrt(2)/2)); label("$6$", shift(5,0)*(sqrt(2)/2,-sqrt(2)/2)); label("$8$", shift(5,0)*(-sqrt(2)/2,-sqrt(2)/2)); label("$1$", shift(10,0)*(-0.5,1)); label("$3$", shift(10,0)*(0.5,1)); label("$5$", shift(10,0)*(1,0)); label("$7$", shift(10,0)*(0.5,-1)); label("$9$", shift(10,0)*(-0.5,-1)); label("$11$", shift(10,0)*(-1,0));[/asy] $ \textbf{(A)}\ \dfrac{1}{4} \qquad \textbf{(B)}\ \dfrac{1}{3} \qquad \textbf{(C)}\ \dfrac{1}{2} \qquad \textbf{(D)}\ \dfrac{2}{3} \qquad \textbf{(E)}\ \dfrac{3}{4}$

It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$. (a) Find $$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$ (b) For which $\beta$ $(1<\beta<2)$ does the following limit exist? $$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$

$P\left(x\right)$ is a polynomial of degree at most $6$ such that such that $P\left(1\right)$, $P\left(2\right)$, $P\left(3\right)$, $P\left(4\right)$, $P\left(5\right)$, $P\left(6\right)$, and $P\left(7\right)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P\left(8\right)$? [i]2018 CCA Math Bonanza Individual Round #13[/i]

Given a fixed positive integer $a\geq 9$. Prove: There exist finitely many positive integers $n$, satisfying: (1)$\tau (n)=a$ (2)$n|\phi (n)+\sigma (n)$ Note: For positive integer $n$, $\tau (n)$ is the number of positive divisors of $n$, $\phi (n)$ is the number of positive integers $\leq n$ and relatively prime with $n$, $\sigma (n)$ is the sum of positive divisors of $n$.

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and $10$ triangles. Find the maximum number of triangles with the same area that may exist in the division.

Each square of a $5 \times 5$ board is painted red or blue, in such a way that the following condition is fulfilled: “For any two rows and two columns, of the $4$ squares that are in their intersections, there are $4$, $2$ or $0$ painted red.” How many ways can the board be painted?

Let $n$ be a positive integer and $x$ positive real number such that none of numbers $x,2x,\dots,nx$ and none of $\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x}$ is an integer. Prove that \[ \left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor \]

We define the distance between two circles $\omega ,\omega '$by the length of the common external tangent of the circles and show it by $d(\omega , \omega ')$. If two circles doesn't have a common external tangent then the distance between them is undefined. A point is also a circle with radius $0$ and the distance between two cirlces can be zero. (a) [b]Centroid.[/b] $n$ circles $\omega_1,\dots, \omega_n$ are fixed on the plane. Prove that there exists a unique circle $\overline \omega$ such that for each circle $\omega$ on the plane the square of distance between $\omega$ and $\overline \omega$ minus the sum of squares of distances of $\omega$ from each of the $\omega_i$s $1\leq i \leq n$ is constant, in other words:\[d(\omega,\overline \omega)^2-\frac{1}{n}{\sum_{i=1}}^n d(\omega_i,\omega)^2= constant\] (b) [b]Perpendicular Bisector.[/b] Suppose that the circle $\omega$ has the same distance from $\omega_1,\omega_2$. Consider $\omega_3$ a circle tangent to both of the common external tangents of $\omega_1,\omega_2$. Prove that the distance of $\omega$ from centroid of $\omega_1 , \omega_2$ is not more than the distance of $\omega$ and $\omega_3$. (If the distances are all defined) (c) [b]Circumcentre.[/b] Let $C$ be the set of all circles that each of them has the same distance from fixed circles $\omega_1,\omega_2,\omega_3$. Prove that there exists a point on the plane which is the external homothety center of each two elements of $C$. (d) [b]Regular Tetrahedron.[/b] Does there exist 4 circles on the plane which the distance between each two of them equals to $1$? Time allowed for this problem was 150 minutes.

求最小的 $n,$ 使得对任意有 ${1000}$ 个顶点且每个顶点度均为 ${4}$ 的简单图 $G,$ 都一定可以从中取掉 ${n}$ 条边$,$ 使 ${G}$ 变为二部图$.$

Consider the parabola $C: y^{2}=4 x$ and the straight line $L: y=x+2$. Let $P$ be a variable point on $L$. Draw the two tangents from $P$ to $C$ and let $Q_{1}$ and $Q_{2}$ denote the two points of contact on $C$. Let $Q$ be the mid-point of the line segment joining $Q_{1}$ and $Q_{2}$. Find the locus of $Q$ as $P$ moves along $L$.

Let $f(x) = \sqrt{4x^2 - 4x^4}$. Let $A$ be the number of real numbers $x$ that satisfy $$f(f(f(\dots f(x)\dots ))) = x,$$ where the function $f$ is applied to $x$ 2020 times. Compute $A \pmod {1000}$. [i]Proposed by Timothy Qian[/i]

(Four Number Theorem) Let $a, b, c,$ and $d$ be positive integers such that $ab=cd$. Show that there exists positive integers $p, q, r,s$ such that \[a=pq, \;\; b=rs, \;\; c=ps, \;\; d=qr.\]

All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron.

Prove that no matter how we number the vertices of a cube with integers from 1 to 8, there exists two opposite vertices in the cube (e.g. they are the endpoints of a large diagonal of the cube), united through a broken line formed with 3 edges of the cube, such that the sum of the 4 numbers written in the vertices of this broken lines is at least 21.

In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.