The interval $[0,1]$ is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least $1/2$.

Given an integer $n \ge 2$, evaluate $\Sigma \frac{1}{pq}$ ,where the summation is over all coprime integers $p$ and $q$ such that $1 \le p < q \le n$ and $p + q > n$.

$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.

Line segments $m$ and $n$ both have length $2$ and bisect each other at an angle of $60^o$, as shown. A point $X$ is placed at uniform random position along $n$, and a point $Y$ is placed at a uniform random position along $m$. Find the probability that the distance between $X$ and $Y$ is less than $\frac12$.

Let $a,b \in [0,1], c \in [-1,1]$ be reals chosen independently and uniformly at random. What is the probability that $p(x) = ax^2+bx+c$ has a root in $[0,1]$?

In a class of $12$ students, no two people are the same height. Compute the total number of ways for the students to arrange themselves in a line such that: [list] [*] for all $1 < i < 12$, the person in the $i$-th position (with the leftmost position being $1$) is taller than exactly $i\pmod 3$ of their adjacent neighbors, and [*] the students standing at positions which are multiples of $3$ are strictly increasing in height from left to right. [/list] [i]Proposed by Nancy Kuang[/i]

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $\$105$, Dorothy paid $\$125$, and Sammy paid $\$175$. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 $

Let $(a_1,a_2,\dots)$ be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets . Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .

Composite numbers $a$ and $b$ have equal number of divisors. All proper divisors of $a$ were written in ascending order and all proper divisors of $b$ were written under them in ascending order, then the numbers that are below each other were added together. It turned out that the resulting numbers formed a set of all proper divisors of a certain number. What are the smallest values that $a$ and $b$ take?

In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a loss is worth $0$ points. Determine the smallest positive integer $n$ for which it is possible that after the $n$-th match all teams have a different number of points and each team has a non-zero number of points.

Let $p = 2027$ be the smallest prime greater than $2018$, and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$. Let $\mathrm{GF}(p)$ be the integers modulo $p$, and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p$). That is, $\mathrm{GF}(p)(X)$ is the set of fractions $\frac{P(X)}{Q(X)}$ of polynomials with coefficients in $\mathrm{GF}(p)$, where $Q(X)$ is not the zero polynomial. Let $D\colon \mathrm{GF}(p)(X)\to \mathrm{GF}(p)(X)$ be a function satisfying \[ D\left(\frac fg\right) = \frac{D(f)\cdot g - f\cdot D(g)}{g^2} \]for any $f,g\in \mathrm{GF}(p)(X)$ with $g\neq 0$, and such that for any nonconstant polynomial $f$, $D(f)$ is a polynomial with degree less than that of $f$. If the number of possible values of $D(P(X))$ can be written as $a^b$, where $a$, $b$ are positive integers with $a$ minimized, compute $ab$. [i]Proposed by Brandon Wang[/i]

A positive integer $n (\ge 4)$ is given. Let $a_1, a_2, \cdots ,a_n$ be $n$ pairwise distinct positive integers where $a_i \le n$ for all $1 \le i \le n$. Determine the maximum value of $$\sum_{i=1}^{n}{|a_i - a_{i+1} + a_{i+2} - a_{i+3}|}$$ where all indices are modulo $n$

A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset". The probability that 10 minutes pass without either baby crying can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #7[/i]

Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and: \[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?

a conic with apotheme 1 slides (varying height and radius, with $r < \frac12$) so that the conic's area is $9$ times that of its inscribed sphere. What's the height of that conic?

Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation: \[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]

The polynomial $P(X)$ is defined by $P(X)=(X+2X^{2}+\ldots +nX^{n})^{2}=a_{0}+a_{1}X+\ldots +a_{2n}X^{2n}$. Prove that $a_{n+1}+a_{n+2}+\ldots +a_{2n}=\frac{n(n+1)(5n^{2}+5n+2)}{24}$.

Three boys agree to divide a bag of marbles in the following manner. The first boy takes one more than half the marbles. The second takes a third of the number remaining. The third boy finds that he is left with twice as many marbles as the second boy. The original number of marbles: $ \textbf{(A)}\ \text{is none of the following} \qquad \textbf{(B)}\ \text{cannot be determined from the given data}\\ \textbf{(C)}\ \text{is 20 or 26} \qquad \textbf{(D)}\ \text{is 14 or 32} \qquad \textbf{(E)}\ \text{is 8 or 38}$

Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying $$P(x + Q(y)) = Q(x + P(y))$$ for all real numbers $x$ and $y$.

Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.

Given is a triangle $ABC$ and the points $M, P$ lie on the segments $AB, BC$, respectively, such that $AM=BC$ and $CP=BM$. If $AP$ and $CM$ meet at $O$ and $2\angle AOM=\angle ABC$, find the measure of $\angle ABC$.

Let $a_1, a_2, \cdots, a_k$ be natural numbers. Let $S(n)$ be the number of solutions in nonnegative integers to $a_1x_1 + a_2x_2 + \cdots + a_kx_k = n$. Suppose $S(n) \neq 0$ for all big enough $n$. Show that for all sufficiently large $n$, we have $S(n+1) < 2S(n)$.

Let $k \in \mathbb{N},$ $S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.$ Any two elements $(a, b)$, $(c, d)$ $\in S_k$ are called "undistinguishing" in $S_k$ if $a - c \equiv 0$ or $\pm 1 \pmod{k}$ and $b - d \equiv 0$ or $\pm 1 \pmod{k}$; otherwise, we call them "distinguishing". For example, $(1, 1)$ and $(2, 5)$ are undistinguishing in $S_5$. Considering the subset $A$ of $S_k$ such that the elements of $A$ are pairwise distinguishing. Let $r_k$ be the maximum possible number of elements of $A$. (i) Find $r_5$. (ii) Find $r_7$. (iii) Find $r_k$ for $k \in \mathbb{N}$.

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$