Show that there exist two consecutive squares such that there are at least $1000$ primes between them.

Tracy has been baking a rectangular cake whose surface is dissected by grid lines in square fields. The number of rows is $ 2^n$ and the number of columns is $ 2^{n \plus{} 1}$ where $ n \geq 1, n \in \mathbb{N}.$ Now she covers the fields with strawberries such that each row has at least $ 2n \plus{} 2$ of them. Show that there four pairwise distinct strawberries $ A,B,C$ and $ D$ which satisfy those three conditions: (a) Strawberries $ A$ and $ B$ lie in the same row and $ A$ further left than $ B.$ Similarly $ D$ lies in the same row as $ C$ but further left. (b) Strawberries $ B$ and $ C$ lie in the same column. (c) Strawberries $ A$ lies further up and further left than $ D.$

Let $m>1$ be a natural number and $N=m^{2017}+1$. On a blackboard, left to right, are written the following numbers: \[N, N-m, N-2m,\dots, 2m+1,m+1, 1.\] On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted. Which numbers will be deleted on the last move.

Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$ In which cases do we have equality?

The large square in the diagram below with sides of length $8$ is divided into $16$ congruent squares. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/6/e/cf828197aa2585f5eab2320a43b80616072135.png[/img]

Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$

How many integers between $80$ and $100$ are prime? $\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.

Let $F$ be a field whose characteristic is not $2$, let $F^*=F\setminus\left\{0\right\}$ be its multiplicative group and let $T$ be the subgroup of $F^*$ constituted by its finite order elements. Prove that if $T$ is finite, then $T$ is cyclic and its order is even.

There are 6 people at a party. Prove that one can [b]either[/b] find a group of $3$ people in which each person is friend with the other two, [b]or[/b] one can find a group of $3$ people in which no two people are friends.

A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$. How many caterpillars of area $n=10^{6}+1$ are there? Proposed by Nikolai Beluhov, Bulgaria

Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

Find all triplets of positive rational numbers $(m,n,p)$ such that the numbers $m+\frac 1{np}$, $n+\frac 1{pm}$, $p+\frac 1{mn}$ are integers. [i]Valentin Vornicu, Romania[/i]

Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.

Let $I$ be the incenter of the triangle $ABC$, and $P$ be any point on the arc $BAC$ of its circumcircle. Points $K$ and $L$ are chosen on the tangent to the circumcircle $\omega$ of triangle $API$ at point $I$, so that $BK = KI$ and $CL = LI$. Show that the circumcircle of triangle $PKL$ is tangent to $\omega$. [i]Proposed by Mykhailo Shtandenko[/i]

Joshua's physics teacher, Dr. Lisi, lives next door to the Kubiks and is a long time friend of the family. An unusual fellow, Dr. Lisi spends as much time surfing and raising chickens as he does trying to map out a $\textit{Theory of Everything}$. Dr. Lisi often poses problems to the Kubik children to challenge them to think a little deeper about math and science. One day while discussing sequences with Joshua, Dr. Lisi writes out the first $2008$ terms of an arithmetic progression that begins $-1776,-1765,-1754,\ldots.$ Joshua then computes the (positive) difference between the $1980^\text{th}$ term in the sequence, and the $1977^\text{th}$ term in the sequence. What number does Joshua compute?

Find all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(1) \neq f(-1)$ and $$f(m+n)^2 \mid f(m)-f(n)$$ for all integers $m, n$. [i]Proposed by Liam Baker, South Africa[/i]

(a) Prove that given constants $a,b$ with $1<a<2<b$, there is no partition of the set of positive integers into two subsets $A_0$ and $A_1$ such that: if $j \in \{0,1\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$. (b) Find all pairs of real numbers $(a,b)$ with $1<a<2<b$ for which the following property holds: there exists a partition of the set of positive integers into three subsets $A_0, A_1, A_2$ such that if $j \in \{0,1,2\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.

[b]p1.[/b] What is the largest number of five dollar footlongs Jimmy can buy with 88 dollars? [b]p2.[/b] Austin, Derwin, and Sylvia are deciding on roles for BMT $2021$. There must be a single Tournament Director and a single Head Problem Writer, but one person cannot take on both roles. In how many ways can the roles be assigned to Austin, Derwin, and Sylvia? [b]p3.[/b] Sofia has$ 7$ unique shirts. How many ways can she place $2$ shirts into a suitcase, where the order in which Sofia places the shirts into the suitcase does not matter? [b]p4.[/b] Compute the sum of the prime factors of $2021$. [b]p5.[/b] A sphere has volume $36\pi$ cubic feet. If its radius increases by $100\%$, then its volume increases by $a\pi$ cubic feet. Compute $a$. [b]p6.[/b] The full price of a movie ticket is $\$10$, but a matinee ticket to the same movie costs only $70\%$ of the full price. If $30\%$ of the tickets sold for the movie are matinee tickets, and the total revenue from movie tickets is $\$1001$, compute the total number of tickets sold. [b]p7.[/b] Anisa rolls a fair six-sided die twice. The probability that the value Anisa rolls the second time is greater than or equal to the value Anisa rolls the first time can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p8.[/b] Square $ABCD$ has side length $AB = 6$. Let point $E$ be the midpoint of $\overline{BC}$. Line segments $\overline{AC}$ and $\overline{DE}$ intersect at point $F$. Compute the area of quadrilateral ABEF. [b]p9.[/b] Justine has a large bag of candy. She splits the candy equally between herself and her $4$ friends, but she needs to discard three candies before dividing so that everyone gets an equal number of candies. Justine then splits her share of the candy between herself and her two siblings, but she needs to discard one candy before dividing so that she and her siblings get an equal number of candies. If Justine had instead split all of the candy that was originally in the large bag between herself and $14$ of her classmates, what is the fewest number of candies that she would need to discard before dividing so that Justine and her $14$ classmates get an equal number of candies? [b]p10.[/b] For some positive integers $a$ and $b$, $a^2 - b^2 = 400$. If $a$ is even, compute $a$. [b]p11.[/b] Let $ABCDEFGHIJKL$ be the equilateral dodecagon shown below, and each angle is either $90^o$ or $270^o$. Let $M$ be the midpoint of $\overline{CD}$, and suppose $\overline{HM}$ splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/387bcdf2a6c39fcada4f21f24ceebd18a7f887.png[/img] [b]p12.[/b] Nelson, who never studies for tests, takes several tests in his math class. Each test has a passing score of $60/100$. Since Nelson's test average is at least $60/100$, he manages to pass the class. If only nonnegative integer scores are attainable on each test, and Nelson gets a dierent score on every test, compute the largest possible ratio of tests failed to tests passed. Assume that for each test, Nelson either passes it or fails it, and the maximum possible score for each test is 100. [b]p13.[/b] For each positive integer $n$, let $f(n) = \frac{n}{n+1} + \frac{n+1}{n}$ . Then $f(1)+f(2)+f(3)+...+f(10)$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p14.[/b] Triangle $\vartriangle ABC$ has point $D$ lying on line segment $\overline{BC}$ between $B$ and $C$ such that triangle $\vartriangle ABD$ is equilateral. If the area of triangle $\vartriangle ADC$ is $\frac14$ the area of triangle $\vartriangle ABC$, then $\left( \frac{AC}{AB}\right)^2$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p15.[/b] In hexagon $ABCDEF$, $AB = 60$, $AF = 40$, $EF = 20$, $DE = 20$, and each pair of adjacent edges are perpendicular to each other, as shown in the below diagram. The probability that a random point inside hexagon $ABCDEF$ is at least $20\sqrt2$ units away from point $D$ can be expressed in the form $\frac{a-b\pi}{c}$ , where $a$, $b$, $c$ are positive integers such that gcd$(a, b, c) = 1$. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/1b45470265d10a73de7b83eff1d3e3087d6456.png[/img] [b]p16.[/b] The equation $\sqrt{x} +\sqrt{20-x} =\sqrt{20 + 20x - x^2}$ has $4$ distinct real solutions, $x_1$, $x_2$, $x_3$, and $x_4$. Compute $x_1 + x_2 + x_3 + x_4$. [b]p17.[/b] How many distinct words with letters chosen from $B, M, T$ have exactly $12$ distinct permutations, given that the words can be of any length, and not all the letters need to be used? For example, the word $BMMT$ has $12$ permutations. Two words are still distinct even if one is a permutation of the other. For example, $BMMT$ is distinct from $TMMB$. [b]p18.[/b] We call a positive integer binary-okay if at least half of the digits in its binary (base $2$) representation are $1$'s, but no two $1$s are consecutive. For example, $10_{10} = 1010_2$ and $5_{10} = 101_2$ are both binary-okay, but $16_{10} = 10000_2$ and $11_{10} = 1011_2$ are not. Compute the number of binary-okay positive integers less than or equal to $2020$ (in base $10$). [b]p19.[/b] A regular octahedron (a polyhedron with $8$ equilateral triangles) has side length $2$. An ant starts on the center of one face, and walks on the surface of the octahedron to the center of the opposite face in as short a path as possible. The square of the distance the ant travels can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/8/3aa6abe02e813095e6991f63fbcf22f2e0431a.png[/img] [b]p20.[/b] The sum of $\frac{1}{a}$ over all positive factors $a$ of the number $360$ can be expressed in the form $\frac{m}{n}$ ,where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the following array: \[ 3, 5\\3, 8, 5\\3, 11, 13, 5\\3, 14, 24, 18, 5\\3, 17, 38, 42, 23, 5\\ \ldots \] Find the 5-th number on the $ n$-th row with $ n>5$.

Let $D$ and $E$ be points on side $[BC]$ of a triangle $ABC$ with circumcenter $O$ such that $D$ is between $B$ and $E$, $|AD|=|DB|=6$, and $|AE|=|EC|=8$. If $I$ is the incenter of triangle $ADE$ and $|AI|=5$, then what is $|IO|$? $ \textbf{(A)}\ \dfrac {29}{5} \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ \dfrac {23}{5} \qquad\textbf{(D)}\ \dfrac {21}{5} \qquad\textbf{(E)}\ \text{None of above} $

Boeotia is comprised of $3$ islands which are home to $2019$ towns in total. Each flight route connects three towns, each on a different island, providing connections between any two of them in both directions. Any two towns in the country are connected by at most one flight route. Find the maximal number of flight routes in the country

A circle $k$ passes through the vertices $B, C$ of a scalene triangle $ABC$. $k$ meets the extensions of $AB, AC$ beyond $B, C$ at $P, Q$ respectively. Let $A_1$ is the foot the altitude drop from $A$ to $BC$. Suppose $A_1P=A_1Q$. Prove that $\widehat{PA_1Q}=2\widehat{BAC}$.

Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?

Let $a_1,a_2,\ldots,a_{2017}$ be the $2017$ distinct complex numbers which satisfy $a_i^{2017}=a_i+1$ for $i=1,2,\ldots,2017$. Compute $$\displaystyle\sum_{i=1}^{2017}\frac{a_i}{a_i^2+1}.$$ [i]2017 CCA Math Bonanza Individual Round #12[/i]