Found problems: 85335
Consider a $10 \times 10$ grid. On every move, we colour $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?
Let $a,b,n$ positive integers with $a>b$ and $ab-1=n^2$. Prove that $a-b \geq \sqrt{4n-3}$ and study the cases where the equality holds.
A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.
Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:
\[AL+BM+CN \leq 3(AD+BE+CF)\]
When does equality occur?
Let $C_1$ be an arbitrary point on the side $AB$ of triangle $ABC$. Points $A_1$ and $B_1$ on the rays $BC$ and $AC$ are such that $\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB$. The lines $AA_1$ and $BB_1$ meet in point $C_2$. Prove that all the lines $C_1C_2$ have a common point.
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
[u]Round 1[/u]
[b]p1.[/b] Five girls and three boys are sitting in a room. Suppose that four of the children live in California. Determine the maximum possible number of girls that could live somewhere outside California.
[b]p2.[/b] A $4$-meter long stick is rotated $60^o$ about a point on the stick $1$ meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters.
[b]p3.[/b] Let $f(x) = 2x(x - 1)^2 + x^3(x - 2)^2 + 10(x - 1)^3(x - 2)$. Compute $f(0) + f(1) + f(2)$.
[u]Round 2[/u]
[b]p4.[/b] Twenty boxes with weights $10, 20, 30, ... , 200$ pounds are given. One hand is needed to lift a box for every $10$ pounds it weighs. For example, a $40$ pound box needs four hands to be lifted. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time.
[b]p5.[/b] Let $ABC$ be a right triangle with a right angle at $A$, and let $D$ be the foot of the perpendicular from vertex$ A$ to side $BC$. If $AB = 5$ and $BC = 7$, compute the length of segment $AD$.
[b]p6.[/b] There are two circular ant holes in the coordinate plane. One has center $(0, 0)$ and radius $3$, and the other has center $(20, 21)$ and radius $5$. Albert wants to cover both of them completely with a circular bowl. Determine the minimum possible radius of the circular bowl.
[u]Round 3[/u]
[b]p7.[/b] A line of slope $-4$ forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle.
[b]p8.[/b] Let $ABC$ be a right triangle with a right angle at $B$, $AB = 9$, and $BC = 7$. Suppose that point $P$ lies on segment $AB$ with $AP = 3$ and that point $Q$ lies on ray $BC$ with $BQ = 11$. Let segments $AC$ and $P Q$ intersect at point $X$. Compute the positive difference between the areas of triangles $AP X$ and $CQX$.
[b]p9.[/b] Fresh Mann and Sophy Moore are racing each other in a river. Fresh Mann swims downstream, while Sophy Moore swims $\frac12$ mile upstream and then travels downstream in a boat. They start at the same time, and they reach the finish line 1 mile downstream of the starting point simultaneously. If Fresh Mann and Sophy Moore both swim at $1$ mile per hour in still water and the boat travels at 10 miles per hour in still water, find the speed of the current.
[u]Round 4[/u]
[b]p10.[/b] The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. The first few terms of the Fibonacci sequence are $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$. Every positive integer can be expressed as the sum of nonconsecutive, distinct, positive Fibonacci numbers, for example, $7 = 5 + 2$. Express $121$ as the sum of nonconsecutive, distinct, positive Fibonacci numbers. (It is not permitted to use both a $2$ and a $1$ in the expression.)
[b]p11.[/b] There is a rectangular box of surface area $44$ whose space diagonals have length $10$. Find the sum of the lengths of all the edges of the box.
[b]p12.[/b] Let $ABC$ be an acute triangle, and let $D$ and $E$ be the feet of the altitudes to $BC$ and $CA$, respectively. Suppose that segments $AD$ and $BE$ intersect at point $H$ with $AH = 20$ and $HD = 13$. Compute $BD \cdot CD$.
PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c4h2809420p24782524]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$
Given triangle $ ABC$. Points $ M$, $ N$ are the projections of $ B$ and $ C$ to the bisectors of angles $ C$ and $ B$ respectively. Prove that line $ MN$ intersects sides $ AC$ and $ AB$ in their points of contact with the incircle of $ ABC$.
In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal.
[i]H. Lesov[/i]
Let $\omega$ be a circle and $C$ a point outside it; distinct points $A$ and $B$ are selected on $\omega$ so that $\overline{CA}$ and $\overline{CB}$ are tangent to $\omega$. Let $X$ be the reflection of $A$ across the point $B$, and denote by $\gamma$ the circumcircle of triangle $BXC$. Suppose $\gamma$ and $\omega$ meet at $D \neq B$ and line $CD$ intersects $\omega$ at $E \neq D$. Prove that line $EX$ is tangent to the circle $\gamma$.
[i]Proposed by David Stoner[/i]
Show that it is impossible to find a triangle in the plane with all integer coordinates such that the lengths of the sides are all odd.
Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set $A$ can have.
In trapezoid $ ABCD$ with $ \overline{BC}\parallel\overline{AD}$, let $ BC\equal{}1000$ and $ AD\equal{}2008$. Let $ \angle A\equal{}37^\circ$, $ \angle D\equal{}53^\circ$, and $ m$ and $ n$ be the midpoints of $ \overline{BC}$ and $ \overline{AD}$, respectively. Find the length $ MN$.
Jongol and Gongol are writing calculus questions and grading tests. They want to write 90 calculus problems and they have 120 tests to grade. Jongol can write 3 questions per minute or grade 4 tests per minute. Gongol can write 1 question per minute or grade 2 tests per minute. Evaluate the shortest possible time, in minutes, for them to complete the two tasks.
[i]2022 CCA Math Bonanza Lightning Round 1.4[/i]
A rhombus is given with one diagonal twice the length of the other diagonal. Express the side of the rhombus is terms of $K$, where $K$ is the area of the rhombus in square inches.
${{ \textbf{(A)}\ \sqrt{K} \qquad\textbf{(B)}\ \frac{1}{2}\sqrt{2K} \qquad\textbf{(C)}\ \frac{1}{3}\sqrt{3K} \qquad\textbf{(D)}\ \frac{1}{4}\sqrt{4K} }\qquad\textbf{(E)}\ \text{None of these are correct} } $
In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row).
(a) Show that $f(C) \in \mathcal F$.
(b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.
What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits $2$ through $9$?
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
Let $ABC$ be a triangle not being isosceles at $A$. Let $(O)$ and $(I)$ denote the circumcircle and incircle of the triangle. $(I)$ touches $AC$ and $AB$ at $E, F$ respectively. Points $M$ and $N$ are on the circle $(I)$ such that $EM \parallel FN \parallel BC$. Let $P,Q$ be the intersections of $BM,CN$ and $(I)$. Prove that
i) $BC,EP, FQ$ are concurrent, and denote by $K$ the point of concurrency.
ii) the circumcircles of triangle $BPK, CQK$ are all tangent to $(I)$ and all pass through a common point on the circle $(O)$.
Nguyễn Minh Hà
Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$.
Determine the largest mediocre number.
We say that a positive integer $k$ is [i]fair [/i] if the number of $2021$-digit palindromes that are a multiple of $k$ is the same as the number of $2022$-digit palindromes that are a multiple of $k$. Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair?
(A palindrome is an integer that reads the same forward and backward.)
[i](David Hruska)[/i]
Let's call the number string $D = d_{n-1}d_{n-2}...d_0$ a [i]stable ending[/i] of a number , if for any natural number $m$ that ends in $D$, any of its natural powers $m^k$ also ends in $D$. Prove that for every natural number $n$ there are exactly four stable endings of a number of length $n$.
[hide=original wording]Ciparu virkni $D = d_{n-1}d_{n-2}...d_0$ sauksim par stabilu skaitļa nobeigumu, ja jebkuram naturālam skaitlim m, kas beidzas ar D, arī jebkura tā naturāla pakāpe $m^k$ beidzas ar D. Pierādīt, ka katram naturālam n ir tieši četri stabili skaitļa nobeigumi, kuru garums ir n.[/hide]
Let $F$ be a finite field having an odd number $m$ of elements. Let $p(x)$ be an irreducible (i.e. nonfactorable) polynomial over $F$ of the form $$x^2+bx+c, ~~~~~~ b,c \in F.$$ For how many elements $k$ in $F$ is $p(x)+k$ irreducible over $F$?