Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

Let $n\ge 2$ be positive integer. Find the least possible number of elements of tile set $A =\{1,2,...,2n-1,2n\}$ that should be deleted in order to the sum of any two different elements remained be a composite number.

How many subsets of $\{1, 2, \cdots, 10\}$ are there that don't contain $2$ consecutive integers?

Find the last eight digits of the binary development of $27^{1986}.$

$ABCD$ is a square and $AB = 1$. Equilateral triangles $AYB$ and $CXD$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $XY$?

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and \[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \] for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.

In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?

On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m$,$n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.

Point $D$ lies on side $AB$ of triangle $ABC$. Let $\omega_1$ and $\Omega_1,\omega_2$ and $\Omega_2$ be the incircles and the excircles (touching segment $AB$) of triangles $ACD$ and $BCD.$ Prove that the common external tangents to $\omega_1$ and $\omega_2,$ $\Omega_1$ and $\Omega_2$ meet on $AB$.

Consider the points $P$ =$(0,0)$,$Q$ = $(1,0)$, $R$= $(2,0)$, $S$ =$(3,0)$ in the $xy$-plane. Let $A,B,C,D$ be four finite sets of colours(not necessarily distinct nor disjoint). In how many ways can $P,Q,R$ be coloured bu colours in $A,B,C$ respectively if adjacent points have to get different colours? In how many ways can $P,Q,R,S$ be coloured by colours in $A,B,C,D$ respectively if adjacent points have to get different colors?

Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$. At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were $252$ handshakes. How many students were in the classroom?

Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n \equal{} 0}^\infty$ such that $ a_n \equal{} 2^n \plus{} a$. Prove that the sequence $ \{a_n\}_{n \equal{} 0}^\infty$ has finitely number of square of integer.

Let $\mathbb{A}$ be the least subset of finite sequences of nonnegative integers that satisfies the following two properties: -$(0,0) \in \mathbb{A}$ - If $(a_1,\ldots,a_n)\in \mathbb{A}$ then $(a_1,\ldots,a_{i-2},a_{i-1}+1,1,a_{i}+1,a_{i+1},\ldots,a_n)\in \mathbb{A}$ for all $i\in \{2,\ldots,n\}$. For each $n\geq 2$, let $\mathbb{B}(n)$ be the set of sequences in $\mathbb{A}$ with $n$ terms. Find the number of elements of $\mathbb{B}$.

Let $n$ be an integer greater than 2, and $P_1, P_2, \cdots , P_n$ distinct points in the plane. Let $\mathcal S$ denote the union of all segments $P_1P_2, P_2P_3, \dots , P_{n-1}P_{n}$. Determine if it is always possible to find points $A$ and $B$ in $\mathcal S$ such that $P_1P_n \parallel AB$ (segment $AB$ can lie on line $P_1P_n$) and $P_1P_n = kAB$, where (1) $k = 2.5$; (2) $k = 3$.

Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.

A polynomial $w$ of degree $n > 1$ has $n$ distinct zeros $x_1,x_2,...,x_n$. Prove that: $$\frac{1}{w'(x_1)}+\frac{1}{w'(x_2)}+...···+\frac{1}{w'(x_n)}= 0.$$

Alex, Bertha, Cameron, Dylan, and Ellen each have a different toy. Each kid puts each of their own toys into a large bag. The toys are then randomly distributed such that each kid receives a toy. How many ways are there for exactly one kid to get the same toy that they put in? [i]2018 CCA Math Bonanza Lightning Round #2.4[/i]

If $r$ is a number such that $r^2-6r+5=0$, find $(r-3)^2$

Bobo the clown was juggling his spherical cows again when he realized that when he drops a cow is related to how many cows he started off juggling. If he juggles $1$, he drops it after $64$ seconds. When juggling $2$, he drops one after $55$ seconds, and the other $55$ seconds later. In fact, he was able to create the following table: [img]https://cdn.artofproblemsolving.com/attachments/1/0/554a9bace83b4b3595c6012dfdb42409465829.png[/img] He can only juggle up to $22$ cows. To juggle the cows the longest, what number of cows should he start off juggling? How long (in minutes) can he juggle for?

Given a positive integer $n$, determine the largest integer $M$ satisfying $$\lfloor \sqrt{a_1}\rfloor + ... + \lfloor \sqrt{a_n} \rfloor \ge \lfloor\sqrt{ a_1 + ... + a_n +M \cdot min(a_1,..., a_n)}\rfloor $$ for all non-negative integers $a_1,...., a_n$. S. Berlov, A. Khrabrov

Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.

[b]p1.[/b] The coins in Merryland all have different integer values: there is a single $1$ cent coin, a single $2$ cent coin, etc. What is the largest number of coins that a resident of Merryland can have if we know that their total value does not exceed $2021$ cents? [b]p2.[/b] For every positive integer $k$ let $$a_k = \left(\sqrt{\frac{k + 1}{k}}+\frac{\sqrt{k+1}}{k}-\frac{1}{k}-\sqrt{\frac{1}{k}}\right).$$ Evaluate the product $a_4a_5...a_{99}$. Your answer must be as simple as possible. [b]p3.[/b] Prove that for every positive integer $n$ there is a permutation $a_1, a_2, . . . , a_n$ of $1, 2, . . . , n$ for which $j + a_j$ is a power of $2$ for every $j = 1, 2, . . . , n$. [b]p4.[/b] Each point of the $3$-dimensional space is colored one of five different colors: blue, green, orange, red, or yellow, and all five colors are used at least once. Show that there exists a plane somewhere in space which contains four points, no two of which have the same color. [b]p5.[/b] Suppose $a_1 < b_1 < a_2 < b_2 <... < a_n < b_n$ are real numbers. Let $C_n$ be the union of $n$ intervals as below: $$C_n = [a_1, b_1] \cup [a_2, b_2] \cup ... \cup [a_n, b_n].$$ We say $C_n$ is minimal if there is a subset $W$ of real numbers $R$ for which both of the following hold: (a) Every real number $r$ can be written as $r = c + w$ for some $c$ in $C_n$ and some $w$ in $W$, and (b) If $D$ is a subset of $C_n$ for which every real number $r$ can be written as $r = d + w$ for some $d$ in $D$ and some $w$ in $W$, then $D = C_n$. (i) Prove that every interval $C_1 = [a_1, b_1]$ is minimal. (ii) Prove that for every positive integer $n$, the set $C_n$ is minimal PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $b$ boys and $g$ girls, with $g \ge 2b-1$, at presence at a party. Each boy invites a girl for the first dance. Prove that this can be done in such a way that either a boy is dancing with a girl he knows or all the girls he knows are not dancing.

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $1999$, then \[|p(0)|\leq C\int_{-1}^1|p(x)|\,dx.\]