Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$. Prove that \[\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},\] where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).

Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.

Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.

In an apartment block there live only couples of parents with children. It is known that every couple has at least one child, that every child has exactly two parents, that every little boy in this building has a sister, and that among the children there are more boys than girls. You may also assume that there are no grandparents living in the building. Is it possible that there are more parents than children in the building? Explain your reasoning.

Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is A. a straight line B. a parabola C. part, but not all, of a parabola D. one branch of a hyperbola E. None of these

Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$. (1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$. (2) Find the value of $ nx_nx_{n \minus{} 1}$. (3) Show that a sequence $ \{x_n\}$ is monotone decreasing. (4) Find $ \lim_{n\to\infty} nx_n^2$.

[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit? [b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice. [b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side. [b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number? [b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score. [b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$. [b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$. [b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. [b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integer $n$ such that $$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$ holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$

Let $\triangle ABC$ be acute with $\angle BAC = 45^{\circ}$. Let $\overline{AD}$ be an altitude of $\triangle ABC$, let $E$ be the midpoint of $\overline{BC}$, and let $F$ be the midpoint of $\overline{AD}$. Let $O$ be the center of the circumcircle of $\triangle ABC$, let $K$ be the intersection of lines $DO$ and $EF$, and let $L$ be the foot of the perpendicular from $O$ to line $AK$. If $BL = 6$ and $CL = 8$, find $AL^2$. [i]Proposed by [b]Awesome_guy[/b][/i]

Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$ and centroid $G$. The circumcircle $k_1$ of triangle $AGC$ and the circumcircle $k_2$ of triangle $BGC$ intersect $AB$ at $P$ and $Q$, respectively. The perpendiculars from $P$ and $Q$ respectively to $AC$ and $BC$ intersect $k_1$ and $k_2$ at $X$ and $Y$. Determine the value of $\frac{CX \cdot CY}{AB^2}$.

Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$

Prove that it is not possible to cover a $6\times 6$ square board with eighteen $2\times 1$ rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a $6\times 5$ rectangle with fifteen $2\times 1 $ rectangles so that the above condition is fulfilled.

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

a government has decided to help it's people by giving them $n$ coupons for $n$ fundamental things, but because of being unmanaged, the giving of the coupons to the people is random. in each time that a person goes to the office to get a coupon, the office manager gives him one of the $n$ coupons randomly and with the same probability. It's obvious that in this system a person may get a coupon that he had it before. suppose that $X_n$ is the random varieble of the first time that a person gets all of the $n$ coupons. show that $\frac{X_n}{n ln(n)}$ in probability converges to $1$.

Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.

Evan, Larry, and Alex are drawing whales on the whiteboard. Evan draws 10 whales, Larry draws 15 whales, and Alex draws 20 whales. Michelle then starts randomly erasing whales one by one. The probability that she finishes erasing Larry's whales first can be expressed as $\frac{p}{q}$. Compute $p+q$. [i]2022 CCA Math Bonanza Team Round #10[/i]

The real numbers $x,y,z$ are such that $x+y+z=4$ and $0 \le x,y,z \le 2$. Find the minimun value of the expression $$A=\sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}$$.

We are given some similar triangles. Their areas are $1^2,3^2,5^2,\cdots,$ and $49^2$. If the smallest triangle has a perimeter of $4$, what is the sum of all the triangles' perimeters?

Lucky Larry's teacher asked him to substitute numbers for $ a$, $ b$, $ c$, $ d$, and $ e$ in the expression $ a\minus{}(b\minus{}(c\minus{}(d\plus{}e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincedence. The numbers Larry substituted for $ a$, $ b$, $ c$, and $ d$ were $ 1$, $ 2$, $ 3$, and $ 4$, respectively. What number did Larry substitute for $ e$? $ \textbf{(A)}\ \minus{}5\qquad\textbf{(B)}\ \minus{}3\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5$

Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$-algebras $\mathcal F_n$, we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$, and for all $n$, $X_n$ is an $\mathcal F_n$-measurable and integrable random variable). Assume that $$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )$$ Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$. [I. Fazekas]

Let $n$ be a positive integer such that there exists a positive integer that is less than $\sqrt{n}$ and does not divide $n$. Let $(a_1, . . . , a_n)$ be an arbitrary permutation of $1, . . . , n$. Let $a_{i1} < . . . < a_{ik}$ be its maximal increasing subsequence and let $a_{j1} > . . . > a_{jl}$ be its maximal decreasing subsequence. Prove that tuples $(a_{i1}, . . . , a_{ik})$ and $(a_{j1}, . . . , a_{jl} )$ altogether contain at least one number that does not divide $n$.

Let $f$ and $g$ be $2 \pi$-periodic integrable functions such that in some neighborhood of $0$, $g(x) = f(ax)$ with some $a \neq 0$. Prove that the Fourier series of $f$ and $g$ are simultaneously convergent or divergent at $0$.

Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions. 1. $f (0) = 2004$. 2. If $x$ is irrational, then $f (x)$ is also irrational. (Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients. A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)

In terms of \(a\), \(b\), and a prime \(p\), find an expression which gives the number of \(x \in \{0, 1, \ldots, p-1\}\) such that the remainder of \(ax\) upon division by \(p\) is less than the remainder of \(bx\) upon division by \(p\).

Let $ABC$ be a right-angled triangle ($\angle B = 90^\circ$). The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1, A_2$ respectively; points $C_1, C_2$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_1C_2, A_1C_1, A_1A_2$ respectively, concur.