Found problems: 4275
A quadratic function has the property that for any interval of length $ 1, $ the length of its image is at least $ 1. $
Show that for any interval of length $ 2, $ the length of its image is at least $ 4. $
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
Find the number of polynomials $P(x)$ of degree $6$ whose coefficients are in the set $\{1,2,\ldots,1999\}$ and which are divisible by $x^3+x^2+x+1$.
[b]p1.[/b] Suppose $\frac{x}{y} = 0.\overline{ab}$ where $x$ and $y$ are relatively prime positive integers and $ab + a + b + 1$ is a multiple of $12$. Find the sum of all possible values of $y$.
[b]p2.[/b] Let $A$ be the set of points $\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}$. How many distinct circles pass through at least three points in $A$?
[b]p3.[/b] Jack and Jill need to bring pails of water home. The river is the $x$-axis, Jack is initially at the point $(-5, 3)$, Jill is initially at the point $(6, 1)$, and their home is at the point $(0, h)$ where $h > 0$. If they take the shortest paths home given that each of them must make a stop at the river, they walk exactly the same total distance. What is $h$?
[b]p4.[/b] What is the largest perfect square which is not a multiple of $10$ and which remains a perfect square if the ones and tens digits are replaced with zeroes?
[b]p5.[/b] In convex polygon $P$, each internal angle measure (in degrees) is a distinct integer. What is the maximum possible number of sides $P$ could have?
[b]p6.[/b] How many polynomials $p(x)$ of degree exactly $3$ with real coefficients satisfy $$p(0), p(1), p(2), p(3) \in \{0, 1, 2\}?$$
[b]p7.[/b] Six spheres, each with radius $4$, are resting on the ground. Their centers form a regular hexagon, and adjacent spheres are tangent. A seventh sphere, with radius $13$, rests on top of and is tangent to all six of these spheres. How high above the ground is the center of the seventh sphere?
[b]p8.[/b] You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules?
[b]p9.[/b] Quadrilateral $ABCD$ has $\overline{AB} = 40$, $\overline{CD} = 10$, $\overline{AD} = \overline{BC}$, $m\angle BAD = 20^o$, and $m \angle ABC = 70^o$. What is the area of quadrilateral $ABCD$?
[b]p10.[/b] We say that a permutation $\sigma$ of the set $\{1, 2,..., n\}$ preserves divisibilty if $\sigma (a)$ divides $\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\{1, 2,..., 40\}$ preserve divisibility? (A permutation of $\{1, 2,..., n\}$ is a function $\sigma$ from $\{1, 2,..., n\}$ to itself such that for any $b \in \{1, 2,..., n\}$, there exists some $a \in \{1, 2,..., n\}$ satisfying $\sigma (a) = b$.)
[b]p11.[/b] In the diagram shown at right, how many ways are there to remove at least one edge so that some circle with an “A” and some circle with a “B” remain connected?
[img]https://cdn.artofproblemsolving.com/attachments/8/7/fde209c63cc23f6d3482009cc6016c7cefc868.png[/img]
[b]p12.[/b] Let $S$ be the set of the $125$ points in three-dimension space of the form $(x, y, z)$ where $x$, $y$, and $z$ are integers between $1$ and $5$, inclusive. A family of snakes lives at the point $(1, 1, 1)$, and one day they decide to move to the point $(5, 5, 5)$. Snakes may slither only in increments of $(1,0,0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Given that at least one snake has slithered through each point of $S$ by the time the entire family has reached $(5, 5, 5)$, what is the smallest number of snakes that could be in the family?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[b]a.)[/b] For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers?
[b]b.)[/b] For which $n>2$ is there exactly one set having this property?
Let $f, g$ be functions from the positive integers to the integers. Vlad the impala is jumping around the integer grid. His initial position is $x_0 = (0, 0)$, and for every $n \ge 1$, his jump is
$x_n - x_{n - 1} = (\pm f(n), \pm g(n))$ or $(\pm g(n), \pm f(n)),$
with eight possibilities in total. Is it always possible that Vlad can choose his jumps to return to his initial location $(0, 0)$ infinitely many times when
(a) $f, g$ are polynomials with integer coefficients?
(b) $f, g$ are any pair of functions from the positive integers to the integers?
Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root.
Proposed by N. Agakhanov
Prove that the polynomial $x^4+\lambda x^3+\mu x^2+\nu x+1$ has no real roots if $\lambda, \mu , \nu $ are real numbers satisfying
\[|\lambda |+|\mu |+|\nu |\le \sqrt{2} \]
Suppose that the closed subset $K$ of the sphere
$$S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}$$
is symmetric with respect to the origin and separates any two antipodal points in $S^2 \backslash K$. Prove that for any positive $\varepsilon$ there exists a homogeneous polynomial $P$ of odd degree such that the Hausdorff distance between
$$Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}$$
and $K$ is less than $\varepsilon$.
Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that
\[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
We are allowed to put several brackets in the expression
$$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$
always in the same places below each other.
(a) Find the smallest possible integer value we can obtain in that way.
(b) Find all possible integer values that can be obtained.
Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.
Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.
Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals:
$\text{(A)}\ \dfrac{5}{2}\qquad
\text{(B)}\ \frac{3}{2}\qquad
\text{(C)}\ \dfrac{4}{3}\qquad
\text{(D)}\ \dfrac{5}{4}\qquad
\text{(E)}\ \dfrac{3}{4}$
Let the polynomials
\[P(x) = x^n + a_{n-1}x^{n-1 }+ \cdots + a_1x + a_0,\]
\[Q(x) = x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0,\]
be given satisfying the identity $P(x)^2 = (x^2 - 1)Q(x)^2 + 1$. Prove the identity
\[P'(x) = nQ(x).\]
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 $y$.
Given a polynomial $P(x,y)$ with real coefficients, suppose that some real function $f:\mathbb R \to \mathbb R$ satisfies
$$P(x,y) = f(x+y)-f(x)-f(y)$$for all $x,y\in\mathbb R$. Show that some polynomial $q$ satisfies
$$P(x,y) = q(x+y)-q(x)-q(y)$$
Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$
[hide=Rules]Time Limit: 25 minutes
Maximum Possible Score: 81
The following is a mathematical Sudoku puzzle which is also a crossword. Your job is to fill in as many blanks as you possibly can, including all shaded squares. You do not earn extra points for showing your work; the only points you get are for correctly filled-in squares. You get one point for each correctly filled-in square. You should read through the following rules carefully before starting.
$\bullet$ Your time limit for this round is $25$ minutes, in addition to the five minutes you get for reading the rules. So make use of your time wisely. The round is based more on speed than on perfect reasoning, so use your intuition well, and be fast.
$\bullet$ This is a Sudoku puzzle; all the squares should be filled in with the digits $1$ through $9$ so that every row and column contains each digit exactly once. In addition, each of the nine $3\times 3$ boxes that compose the grid also contains each digit exactly once. Furthermore, this is a super-Sudoku puzzle; in addition to satisfying all these conditions, the four $3\times 3$ boxes with red outlines also contain each of $1,..., 9$ exactly once. This last property is important to keep in mind – it may help you solve the puzzle faster.
$\bullet$ Just to restate the idea, you can use the digits $1$ through $9$, but not $0$. You may not use any other symbol, such as $\pi$ or $e$ or $\epsilon$. Each square gets exactly one digit.
$\bullet$ The grid is also a crossword puzzle; the usual rules apply. The shaded grey squares are the “black” squares of an ordinary crossword puzzle. The white squares as well as the shaded yellow ones count as the “white” crossword squares. All squares, white or shaded, count as ordinary Sudoku squares.
$\bullet$ If you obtain the unique solution to the crossword puzzle, then this solution extends to a unique solution to the Sudoku puzzle.
$\bullet$ You may use a graphing calculator to help you solve the clues.
The following hints and tips may prove useful while solving the puzzle.
$\bullet$ Use the super-Sudoku structure described in the first rule; use all the symmetries you have. Remember that we are not looking for proofs or methods, only for correctly filled-in squares.
$\bullet$ If you find yourself stuck on a specific clue, it is nothing to worry about. You can obtain the solution to that clue later on by solving other clues and figuring out certain digits of your desired solution. Just move on to the rest of the puzzle.
$\bullet$ As you progress through the puzzle, keep filling in all squares you have found on your solution sheet, including the shaded ones. Remember that for scoring, the shaded grey squares count the same as the white ones.
Good luck!
[/hide]
[asy]
// place label "s" in row i, column j
void labelsq(int i, int j, string s) { label("$"+s+"$",(j-0.5,7.5-i),fontsize(14)); }
// for example, use the command
// labelsq(1,7,"2");
// to put the digit 2 in the top right box
// **** rest of code ****
size(250); defaultpen(linewidth(1));
pair[] labels = {(1,1),(1,4),(1,6),(1,7),(1,9),(2,1),(2,6),(3,4),(4,1),(4,8),(5,1),(6,3),(6,5),(6,6),(7,1),(7,2),(7,7),(7,9),(8,1),(8,4),(9,1),(9,6)};
pair[] blacksq = {(1,5),(2,5),(3,2),(3,3),(3,8),(5,5),(5,6),(5,7),(5,9),(6,2),(6,7),(6,9),(8,3),(9,5),(9,8)};
path peachsq = shift(1,1)*scale(3)*unitsquare; pen peach = rgb(0.98,0.92,0.71); pen darkred = red + linewidth(2);
fill(peachsq,peach); fill(shift(4,0)*peachsq,peach); fill(shift(4,4)*peachsq,peach); fill(shift(0,4)*peachsq,peach);
for(int i = 0; i < blacksq.length; ++i) fill(shift(blacksq[i].y-1, 9-blacksq[i].x)*unitsquare, gray(0.6));
for(int i = 0; i < 10; ++i) { pen sudokuline = linewidth(1); if(i == 3 || i == 6) sudokuline = linewidth(2); draw((0,i)--(9,i),sudokuline); draw((i,0)--(i,9),sudokuline); }
draw(peachsq,darkred); draw(shift(4,0)*peachsq,darkred); draw(shift(4,4)*peachsq,darkred); draw(shift(0,4)*peachsq,darkred);
for(int i = 0; i < labels.length; ++i) label(string(i+1), (labels[i].y-1, 10-labels[i].x), SE, fontsize(10));
// **** draw letters ****
draw(shift(.5,.5)*((0,6)--(0,8)--(2,8)--(2,7)--(0,7)^^(3,8)--(3,6)--(5,6)--(5,8)^^(6,6)--(6,8)--(7,8)--(7,7)--(7,8)--(8,8)--(8,6)^^(0,3)--(0,5)--(2,5)--(2,3)--(2,4)--(0,4)^^(5,3)--(3,3)--(3,5)--(5,5)),linewidth(1)+rgb(0.94,0.74,0.58));
// **** end rest of code ****[/asy]
[b][u][i]Across[/i][/u][/b]
[b]1 Across.[/b] The following is a normal addition where each letter represents a (distinct) digit: $$GOT + TO + GO + TO = TOP$$This certainly does not have a unique solution. However, you discover suddenly that $G = 2$ and $P \notin \{4, 7\}$. Then what is the numeric value of the expression $GOT \times TO$?
[b]3 Across.[/b] A strobogrammatic number which reads the same upside down, e.g. $619$. On the other hand, a triangular number is a number of the form $n(n + 1)/2$ for some $n \in N$, e.g. $15$ (therefore, the $i^{th}$ triangular number $T_i$ is the sum of $1$ through $i$). Let $a$ be the third strobogrammatic prime number. Let $b$ be the smaller number of the smallest pair of triangular numbers whose sum and difference are also triangular numbers. What is the value of $ab$?
[b]6 Across.[/b] A positive integer $m$ is said to be palindromic in base $\ell$ if, when written in base $\ell$ , its digits are the same front-to-back and back-to-front. For $j, k \in N$, let $\mu (j, k)$ be the smallest base-$10$ integer that is palindromic in base $j$ as well as in base$ k$, and let $\nu (j, k) := (j + k) \cdot \mu (j, k)$. Find the value of $\nu (5, 9)$.
[b]7 Across.[/b] Suppose you have the unique solution to this Sudoku puzzle. In that solution, let $X$ denote the sum of all digits in the shaded grey squares. Similarly, let $Y$ denote the sum of all numbers in the shaded yellow squares on the upper left block (i.e. the $3 \times 3$ box outlined red towards the top left). Concatenate $X$ with $Y$ in that order, and write that down.
[b]8 Across.[/b] For any $n \in N$ such that $1 < n < 10$, define the sequence $X_{n,1}$,$X_{n,2}$,$ ...$ by $X_{n,1} = n$, and for $r \ge 2$, X_{n,r} is smallest number $k \in N$ larger than X_{n,r-1} such that $k$ and the sum of digits of $k$ are both powers of $n$. For instance, $X_{3,1 = 3}$, $X_{3,2} = 9$, $X_{3,3} = 27$, and so on. Concatenate $X_{2,2}$ with $X_{2,4}$, and write down the answer.
[b]9 Across.[/b] Find positive integers $x, y,z$ satisfying the following properties: $y$ is obtained by subtracting $93$ from $x$, and $z$ is obtained by subtracting $183$ from $y$, furthermore, $x, y$ and $z$ in their base-$10$ representations contain precisely all the digits from $1$ through $9$ once (i.e. concatenating $x, y$ and $z$ yields a valid $9$-digit Sudoku answer). Obviously, write down the concatenation of $x, y$ and $z$ in that order.
[b]11 Across.[/b] Find the largest pair of two-digit consecutive prime numbers $a$ and $b$ (with $a < b$) such that the sum of the digits of a plus the sum of the digits of b is also a prime number. Write the concatenation of $a$ and $b$.
[b]12 Across.[/b] Suppose you have a strip of $2n + 1$ squares, with n frogs on the $n$ squares on the left, and $n$ toads on the $n$ squares on the right. A move consists either of a toad or a frog sliding to an adjacent square if it is vacant, or of a toad or a frog jumping one square over another one and landing on the next square if it is vacant. For instance, the starting position
[img]https://cdn.artofproblemsolving.com/attachments/a/a/6c97f15304449284dc282ff86014f526322e4a.png[/img]
has the position
[img]https://cdn.artofproblemsolving.com/attachments/e/6/e2c9520731bd94dc0aa37f540c2b9d1bce6432.png[/img]
or the position
[img]https://cdn.artofproblemsolving.com/attachments/3/f/06868eca80d649c4f80425dc9dc5c596cb2a4e.png[/img]
as results of valid first moves. What is the minimum number of moves needed to swap the toads with the frogs if $n = 5$? How about $n = 6$? Concatenate your answers.
[b]15 Across.[/b] Let $w$ be the largest number such that $w$, $2w$ and $3w$ together contain every digit from $1$ through $9$ exactly once. Let $x$ be the smallest integer with the property that its first $5$ multiples contain the digit $9$. A Leyland number is an integer of the form $m^n + n^m$ for integers $m, n > 1$. Let $y$ be the fourth Leyland number. A Pillai prime is a prime number $p$ for which there is an integer $n > 0$ such that $n! \equiv - 1 (mod \,\, p)$, but $p \not\equiv 1 (mod \,\, n)$. Let $z$ be the fourth Pillai prime. Concatenate $w$, $x, y$ and $z$ in that order to obtain a permutation of $1,..., 9$. Write down this permutation.
[b]19 Across.[/b] A hoax number $k \in N$ is one for which the sum of its digits (in base $10$) equals the sum of the digits of its distinct prime factors (in base $10$). For instance, the distinct prime factors of $22$ are $2$ and $11$, and we have $2+2 = 2+(1+1)$. In fact, $22$ is the first hoax number. What is the second?
[b]20 Across.[/b] Let $a, b$ and $c$ be distinct $2$-digit numbers satisfying the following properties:
– $a$ is the largest integer expressible as $a = x^y = y^x$, for distinct integers $x$ and $y$.
– $b$ is the smallest integer which has three partitions into three parts, which all give the same product (which turns out to be $1200$) when multiplied.
– $c$ is the largest number that is the sum of the digits of its cube.
Concatenate $a, b$ and $c$, and write down the resulting 6-digit prime number.
[b]21 Across.[/b] Suppose $N = \underline{a}\, \underline{b} \, \underline{c} \, \underline{d}$ is a $4$-digit number with digits $a, b, c$ and $d$, such that $N = a \cdot b \cdot c \cdot d^7$. Find $N$.
[b]22 Across.[/b] What is the smallest number expressible as the sum of $2, 3, 4$, or $5$ distinct primes?
[b][u][i]Down [/i][/u][/b]
[b]1 Down.[/b] For some $a, b, c \in N$, let the polynomial $$p(x) = x^5 - 252x^4 + ax^3 - bx^2 + cx - 62604360$$ have five distinct roots that are positive integers. Four of these are 2-digit numbers, while the last one is single-digit. Concatenate all five roots in decreasing order, and write down the result.
[b]2 Down.[/b] Gene, Ashwath and Cosmin together have $2511$ math books. Gene now buys as many math books as he already has, and Cosmin sells off half his math books. This leaves them with $2919$ books in total. After this, Ashwath suddenly sells off all his books to buy a private jet, leaving Gene and Cosmin with a total of $2184$ books. How many books did Gene, Ashwath and Cosmin have to begin with? Concatenate the three answers (in the order Gene, Ashwath, Cosmin) and write down the result.
[b]3 Down.[/b] A regular octahedron is a convex polyhedron composed of eight congruent faces, each of which is an equilateral triangle; four of them meet at each vertex. For instance, the following diagram depicts a regular octahedron:
[img]https://cdn.artofproblemsolving.com/attachments/c/1/6a92f12d5e9f56b0699531ae8369a0ab8ab813.png[/img]
Let $T$ be a regular octahedron of edge length $28$. What is the total surface area of $T$ , rounded to the nearest integer?
[b]4 Down.[/b] Evaluate the value of the expression $$\sum^{T_{25}}_{k=T_{24}+1}k, $$ where $T_i$ denotes the $i^{th}$ triangular number (the sum of the integers from $1$ through $i$).
[b]5 Down.[/b] Suppose $r$ and $s$ are consecutive multiples of$ 9$ satisfying the following properties:
– $r$ is the smallest positive integer that can be written as the sum of $3$ positive squares in $3$ different ways.
– $s$ is the smallest $2$-digit number that is a Woodall number as well as a base-$10$ Harshad number. A Woodall number is any number of the form $n \cdot 2^n - 1$ for some $n \in N$. A base-$10$ Harshad number is divisible by the sum of its digits in base $10$.
Concatenate $r$ and $s$ and write down the result.
[b]10 Down.[/b] For any $k \in N$, let $\phi_p(k)$ denote the sum of the distinct prime factors of $k$. Suppose $N$ is the largest integer less than $50000$ satisfying $\phi_p(N) =\phi_p(N + 1)$, where the common value turns out to be a meager $55$. What is$ N$?
[b]13 Down.[/b] The $n^{th}$ $s$-gonal number $P(s, n)$ is defined as $$P(s, n) = (s - 3)T_{n-1} + T_n$$ where $T_i$ is the $i^{th}$ triangular number (recall that the $i^{th}$ triangular number is the sum of the numbers $1$ through $i$). Find the least $N$ such that $N$ is both a $34$-gonal number, and a $163$-gonal number.
[b]14 Down.[/b] A biprime is a positive integer that is the product of precisely two (not necessarily distinct) primes. A cluster of biprimes is an ordered triple $(m,m + 1,m + 2)$ of consecutive integers that are biprimes. There are precisely three clusters of biprimes below 100. Denote these by, say, $$\{(p, p + 1, p + 2), (q, q + 1,q + 2), (r, r + 1, r + 2)\}$$ and add the condition that $p + 2 < q < r - 2$ to fix the three clusters. Interestingly, $p + 1$ and $q$ are both multiples of $17$. Concatenate $q$ with $p + 1$ in that order, and write down the result.
[b]16 Down.[/b] Find the least positive integer $m$ (written in base $10$ as $m = \underline{a} \, \underline{b} \, \underline{c} $, with digits $a, b,c$), such that $m = (b + c)^a$.
[b]17 Down.[/b] Let $X$ be a set containing $32$ elements, and let $Y\subseteq X$ be a subset containing $29$ elements. How many $2$-element subsets of $X$ are there which have nonempty intersection with $Y$?
[b]18 Down.[/b] Find a positive integer $K < 196$, which is a strange twin of the number $196$, in the sense that $K^2$ shares the same digits as $196^2$, and $K^3$ shares the same digits as $196^3$.
PS. You should use hide for answers.
Karina has a polynomial $p_1(x) = x^2 + x + k$, where $k$ is an integer. Noticing that $p_1$ has integer roots, she forms a new polynomial $p_2(x) = x^2 + a_1x + b_1$, where $a_1$ and $b_1$ are the roots of $p_1$ and $a_1 \ge b_1$. The polynomial $p_2$ also has integer roots, so she forms a new polynomial $p_3(x) = x^2 + a_2x + b_2$, where $a_2$ and $b_2$ are the roots of $p_2$ and $a_2 \ge b_2$. She continues this process until she reaches $p_7(x)$ and finds that it does not have integer roots. What is the largest possible value of $k$?
[u]Round 1[/u]
[b]p1.[/b] A $\$100$ TV has its price increased by $10\%$. The new price is then decreased by $10\%$. What is the current price of the TV?
[b]p2.[/b] If $9w + 8x + 7y = 42$ and $w + 2x + 3y = 8$, then what is the value of $100w + 101x + 102y$?
[b]p3.[/b] Find the number of positive factors of $37^3 \cdot 41^3$.
[u]Round 2[/u]
[b]p4.[/b] Three hoses work together to fill up a pool, and each hose expels water at a constant rate. If it takes the first, second, and third hoses 4, 6, and 12 hours, respectively, to fill up the pool alone, then how long will it take to fill up the pool if all three hoses work together?
[b]p5.[/b] A semicircle has radius $1$. A smaller semicircle is inscribed in the larger one such that the two bases are parallel and the arc of the smaller is tangent to the base of the larger. An even smaller semicircle is inscribed in the same manner inside the smaller of the two semicircles, and this procedure continues indefinitely. What is the sum of all of the areas of the semicircles?
[b]p6.[/b] Given that $P(x)$ is a quadratic polynomial with $P(1) = 0$, $P(2) = 0$, and $P(0) = 2012$, find $P(-1)$.
[u]Round 3[/u]
[b]p7.[/b] Darwin has a paper circle. He labels one point on the circumference as $A$. He folds $A$ to every point on the circumference on the circle and undoes it. When he folds $A$ to any point $P$, he makes a blue mark on the point where $\overline{AP}$ and the made crease intersect. If the area of Darwin paper circle is 80, then what is the area of the region surrounded by blue?
[b]p8.[/b] Α rectangular wheel of dimensions $6$ feet by $8$ feet rolls for $28$ feet without sliding. What is the total distance traveled by any corner on the rectangle during this roll?
[b]p9[/b]. How many times in a $24$-hour period do the minute hand and hour hand of a $12$-hour clock form a right angle?
[u]Round 4[/u]
The answers in this section all depend on each other. Find smallest possible solution set.
[b]p10.[/b] Let B be the answer to problem $11$. Right triangle $ACD$ has a right angle at $C$. Squares $ACEF$ and $ADGH$ are drawn such that points $D$ and $E$ do not coincide and points $E$ and $H$ do not coincide. The midpoints of the sides of $ADGH$ are connected to form a smaller square with area $B.$ If the area of $ACEF$ is also $B$, then find the length $CD$ rounded up to the nearest integer.
[b]p11.[/b] Let $C$ be the answer to problem $12$. Find the sum of the digits of $C$.
[b]p12.[/b] Let $A$ be the answer to problem $10$. Given that $a_0 = 1$, $a_1 = 2$, and that $a_n = 3a_{n-1 }-a_{n-2}$ for $n \ge 2$, find $a_A$.
PS. You should use hide for answers.Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.