Evaluate $\tfrac1{\sqrt1+\sqrt2}+\tfrac1{\sqrt2+\sqrt3}+\cdots+\tfrac1{\sqrt{1368}+\sqrt{1369}}$.

Given is a set of $n\ge5$ people and $m$ commissions with $3$ persons in each. Let all the commissions be [i]nice[/i] if there are no two commissions $A$ and $B$, such that $\mid A\cap B\mid=1$. Find the biggest possible $m$ (as a function of $n$).

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$. Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$.

Let $n$ be the answer to this problem. We define the digit sum of a date as the sum of its $4$ digits when expressed in mmdd format (e.g. the digit sum of $13$ May is $0+5+1+3 = 9$). Find the number of dates in the year $2021$ with digit sum equal to the positive integer $n$.

In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.

Given right $n$-gon wit the point $O$ -- its centre. All the vertices are marked either with $+1$ or $-1$. We may change all the signs in the vertices of regular $k$-gon ($2 \le k \le n$) with the same centre $O$. (By $2$-gon we understand a segment, being halved by $O$.) Prove that in a), b) and c) cases there exists such a set of $(+1)$s and $(-1)$s, that we can never obtain a set of $(+1)$s only. a) $n = 15$, b) $n = 30$, c) $n > 2$, d) Let us denote $K(n)$ the maximal number of $(+1)$ and $(-1)$ sets such, that it is impossible to obtain one set from another. Prove, for example, that $K(200) = 2^{80}$

Natural numbers $1,2,...,500$ are written on a blackboard. Two players $A$ and $B$ consecutively make moves, $A$ starts. Each move a player chooses two numbers $n$ and $2n$ and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

In a triangle $ABC$ with $\angle BAC = 90^{\circ}$ let $BL$ be the bisector, $L\in AC$. Let $D$ be a point symmetrical to $A$ with respect to $BL$. Let $M$ be the circumcenter of $ADC$. Prove that $CM$, $DL$, and $AB$ are concurrent.

Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$

GM Bisain's IQ is so high that he can move around in $10$ dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates. [i]2019 CCA Math Bonanza Lightning Round #4.2[/i]

What is the least number of weighings needed to determine the sum of weights of $13$ watermelons such that exactly two watermelons should be weighed in each weigh? $ \textbf{a)}\ 7 \qquad\textbf{b)}\ 8 \qquad\textbf{c)}\ 9 \qquad\textbf{d)}\ 10 \qquad\textbf{e)}\ 11 $

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

Plusses and minuses are inserted in the expression \[\pm 1 \pm 2 \pm 3 \dots \pm 2016\] such that when evaluated the result is divisible by 2017. Let there be $N$ ways for this to occur. Compute the remainder when $N$ is divided by 503. [i]2016 CCA Math Bonanza Team #10[/i]

For a set $A$ of integers, define $f(A)=\{x^2+xy+y^2: x,y\in A\}$. Is there a constant $c$ such that for all positive integers $n$, there exists a set $A$ of size $n$ such that $|f(A)|\le cn$? [i]David Yang.[/i]

In triangle $ABC$ we have $AB = BC, \angle B = 20^o$. Point $M$ on $AC$ is such that $AM : MC = 1 : 2$, point $H$ is the projection of $C$ to $BM$. Find angle $AHB$. (M. Yevdokimov)

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

Let $ABC$ be a triangle inscribed in circle $\Gamma$, centered at $O$ with radius $333.$ Let $M$ be the midpoint of $AB$, $N$ be the midpoint of $AC$, and $D$ be the point where line $AO$ intersects $BC$. Given that lines $MN$ and $BO$ concur on $\Gamma$ and that $BC = 665$, find the length of segment $AD$. [i]Author: Alex Zhu[/i]

Compute the number of solutions to $1+\cos(\theta)+\cos(2\theta)+\ldots+\cos(2024\theta) = \tfrac{1}{2}$ for $\theta \in [0,2\pi].$

Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$

[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$? [b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails? [b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$. [b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$? [b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average. [b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number. [b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle. [b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping. [b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$. [b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Does there exist a natural number $n \geq 2$ such that: a) $\frac{2^{n-1}+1}{n}$ is a natural number? b) $\frac{2^{2n-1}-1}{n}$ is a prime number?

There are $20$ collinear points, separated by the same distance: $$. \,\,\, . \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\,. \,\,\, .$$ Miguel has to paint three or more of these points red, in such a way that the red points are separated by the same distance and it is impossible to paint exactly one more point red without violating the previous condition. Determine in how many ways Miguel can do his homework.

We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence?