If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy $lcm (a, b, c, d) < 1000$ and $a+b = c+d$, compute the largest possible value of $a + b$.

Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .

For all positive real numbers $a_1, a_2, \dots, a_n$, prove that $$ \frac{a_1\! +\! a_2}{2} \cdot \frac{a_2\! +\! a_3}{2} \cdot \dots \cdot \frac{a_n\! +\! a_1}{2} \leq \frac{a_1\!+\!a_2\!+\!a_3}{2 \sqrt{2}} \cdot \frac{a_2\!+\!a_3\!+\!a_4}{2 \sqrt{2}} \cdot \dots \cdot \frac{a_n\!+\!a_1\!+\!a_2}{2 \sqrt{2}}.$$

Let $M=\{1,2,...,n\}$. Prove that the number of pairs $(A,a)$, where $A\subset M$ and $a$ is a permutation of $M$, for which $a(A)\cap A=\emptyset $, is equal to $n!.F_{n+1}$, where $F_{n+1}$ is the $n+1$ member of the Fibonacci sequence.

A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines? $\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$

Given positive integer $n$, find the biggest real number $C$ which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than $1$ is less than $C$, then we can divide the set of numbers into no more than $n$ groups so that the sum of reciprocals of every group is less than $1$.

Say that an ordered triple $(a, b, c)$ is [i]pleasing[/i] if (a) $a$, $b$, and $c$ are in the set $\{ 1, 2, \dots, 17 \}$, and (b) both $b - a$ and $c - b$ are greater than $3$, and at least one of them is equal to $4$. How many pleasing triples are there?

Call an [i]ordered[/i] triple $(a,b,c)$ [i]$d$-tall[/i] if there exists a triangle with side lengths $a,b,c$ and the height to the side with length $a$ is $d$. Suppose that for some positive integer $k$, there are exactly $210$ $k$-tall ordered triples of positive integers. How many $k$-tall ordered triples $(a,b,c)$ are there such that a triangle $ABC$ with $BC=a,CA=b,AB=c$ satisfies both $\angle{B}<90^\circ$ and $\angle{C}<90^\circ$? [i]2020 CCA Math Bonanza Team Round #8[/i]

Let $ABC$ be a triangle, and let $D$ be the foot of the $A-$altitude. Points $P, Q$ are chosen on $BC$ such that $DP = DQ = DA$. Suppose $AP$ and $AQ$ intersect the circumcircle of $ABC$ again at $X$ and $Y$. Prove that the perpendicular bisectors of the lines $PX$, $QY$, and $BC$ are concurrent. [i]Proposed by Pranjal Srivastava[/i]

Let $\alpha$ and $\beta$ be reals. Find the least possible value of $$(2 \cos \alpha + 5 \sin \beta - 8)^2 + (2 \sin \alpha + 5 \cos \beta - 15)^2.$$

Let $ n$ denote the smallest positive integer that is divisible by both $ 4$ and $ 9$, and whose base-$ 10$ representation consists of only $ 4$'s and $ 9$'s, with at least one of each. What are the last four digits of $ n$? $ \textbf{(A)}\ 4444\qquad \textbf{(B)}\ 4494\qquad \textbf{(C)}\ 4944\qquad \textbf{(D)}\ 9444\qquad \textbf{(E)}\ 9944$

Find all positive integers $n$ such that the elements of $$\{1,2,...,2n+1\}-\{n+1\}$$ can be partitioned into two groups with the same number of elements and the same sum of their elements.

Let $a,b,x\in \mathbb{N}$ with $b>1$ and such that $b^{n}-1$ divides $a$. Show that in base $b$, the number $a$ has at least $n$ non-zero digits.

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.

[u]Version for Nordic Countries[/u] Six pine trees grow on the shore of a circular lake. It is known that a treasure is submerged at the mid-point $T$ between the intersection points of the altitudes of two triangles, the vertices of one being at three of the $6$ pines, and the vertices of the second one at the other three pines. At how many points $T$ must one dive to find the treasure? [u]Version for Tropical Countries[/u] A captain finds his way to Treasure Island, which is circular in shape. He knows that there is treasure buried at the midpoint of the segment joining the orthocentres of triangles $ABC$ and $DEF$, where $A$, $B$, $C$, $D$, $E$ and $F$ are six palm trees on the shore of the island, not necessarily in cyclic order. He finds the trees all right, but does not know which tree is denoted by which letter. What is the maximum number of points at which the captain has to dig in order to recover the treasure? (S Markelov)

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

Let $n$ be a positive integer. Alice writes $n$ real numbers $a_1, a_2,\dots, a_n$ in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ($a_n$ is not a neighbor of $a_1$, nor vice versa). A number [i]changes sign[/i] if it changes from being nonnegative to negative or vice versa. In terms of $n$, determine the maximum number of times that $a_1$ can change sign, across all possible values of $a_1,a_2,\dots, a_n$ and all possible sequences of moves Alice may make.

Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse? [b]Remark:[/b] A black horse could attack another black horse.

Let $k$ be a positive integer and let $m=6k-1$. Let $$S(m)=\sum_{j=1}^{2k-1}(-1)^{j+1}\binom m{3j-1}.$$Prove that $S(m)$ is never zero.

On a tournament with $ n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a [i]draw-triple[/i] if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.

$\frac{2^1+2^0+2^{-1}}{2^{-2}+2^{-3}+2^{-4}}$ equals $\text{(A)} \ 6 \qquad \text{(B)} \ 8 \qquad \text{(C)} \ \frac{31}{2} \qquad \text{(D)} \ 24 \qquad \text{(E)} \ 512$

In the diagram of rectangles below, with lengths as labeled, let $A$ be the area of the rectangle labeled $A$, and so on. Find $36A+6B+C+6D$. [asy] size(3cm); real[] A = {0,8,13}; real[] B = {0,7,12}; for (int i = 0; i < 3; ++i) { draw((A[i],0)--(A[i],-B[2])); draw((0,-B[i])--(A[2],-B[i])); } label("8", (4,0), N); label("5", (10.5,0),N); label("7", (0,-3.5),W); label("5", (0,-9.5),W); label("$A$", (4,-3.5)); label("$B$", (10.5,-3.5)); label("$C$", (10.5,- 9.5)); label("$D$", (4, -9.5)); [/asy] [i]2018 CCA Math Bonanza Team Round #1[/i]

The numeral $47$ in base $a$ represents the same number as $74$ in base $b$. Assuming that both bases are positive integers, the least possible value of $a+b$ written as a Roman numeral, is $\textbf{(A) }\mathrm{XIII}\qquad\textbf{(B) }\mathrm{XV}\qquad\textbf{(C) }\mathrm{XXI}\qquad\textbf{(D) }\mathrm{XXIV}\qquad \textbf{(E) }\mathrm{XVI}$

Let $ABC$ be an acute triangle with $AB\le AC$ and let $c(O,R)$ be it's circumscribed circle (with center $O$ and radius $R$). The perpendicular from vertex $A$ on the tangent of the circle passing through point $C$, intersect it at point $D$. a) If the triangle $ABC$ is isosceles with $AB=AC$, prove that $CD=BC/2$. b) If $CD=BC/2$, prove that the triangle $ABC$ is isosceles.

The infinite periodic fractions $\frac{a} {b}$ and $\frac{c} {d}$ with $(a, b)=(c, d)=1$ are such that every finite block of digits in the first fraction after the decimal point appears in the second fraction as well (again after the decimal point). Show that $b=d$.