Found problems: 85335
Let $A,B,C,D,E$ be five different points on the circumference of a circle in that (cyclic) order. Let $F$ be the intersection of chords $BD$ and $CE$. Show that if $AB=AE=AF$ then lines $AF$ and $CD$ are perpendicular.
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and yellow when in the sun. Initially the ratio of green to yellow frogs was 3:1. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is 4:1. What is the difference between the number of green frogs and yellow frogs now?
$\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24$
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.
Let $ABC$ be a triangle such that $\angle CAB > \angle ABC$, and let $I$ be its incentre. Let $D$ be the point on segment $BC$ such that $\angle CAD = \angle ABC$. Let $\omega$ be the circle tangent to $AC$ at $A$ and passing through $I$. Let $X$ be the second point of intersection of $\omega$ and the circumcircle of $ABC$. Prove that the angle bisectors of $\angle DAB$ and $\angle CXB$ intersect at a point on line $BC$.
$M$ is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of $M.$ How many elements can $M$ have at most?
How many permutations $p$ of $\{1, 2, ..., 8\}$ satisfy $|p(p(a)) - a| \leq 1$ for all $a$?
[i]Team #9[/i]
What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA? (For example, 3 swaps are required to change ABC to CBA; one such sequence of swaps is ABC $\rightarrow$ BAC $\rightarrow$ BCA $\rightarrow$ CBA.)
$
\textbf{(A) }6 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }12 \qquad
\textbf{(D) }15 \qquad
\textbf{(E) }24 \qquad
$
In acute triangle $ABC$ the bisector of $\measuredangle A$ meets side $BC$ at $D$. The circle with center $B$ and radius $BD$ intersects side $AB$ at $M$; and the circle with center $C$ and radius $CD$ intersects side $AC$ at $N$. Then it is always true that
$ \textbf{(A)}\ \measuredangle CND+\measuredangle BMD-\measuredangle DAC =120^{\circ} \qquad\textbf{(B)}\ AMDN\ \text{is a trapezoid} \qquad\textbf{(C)}\ BC\ \text{is parallel to}\ MN \\ \qquad\textbf{(D)}\ AM-AN=\frac{3(DB-DC)}{2} \qquad\textbf{(E)}\ AB-AC=\frac{3(DB-DC)}{2}$
Let $n\ge 2$ be a positive integer, and consider an $n\times n$ grid of $n^2$ equilateral triangles. Two triangles are adjacent if they share at least one vertex. Each triangle is colored red or blue, splitting the grid into regions.
Find, with proof, the minimum number of triangles in the largest region.
[i]Rohan Bodke[/i]
The integers from $1$ through $9$ inclusive, are placed in the squares of a $3 \times 3$ grid. Each square contains a different integer. The product of the integers in the first and second rows are $60$ and $96$ respectively. Find the sum of the integers in the third row.
[i]Proposed by bissue [/i]
There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$'s car fits $\chi (p_i)$ people ($\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$). A $c$-fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$. )
Suppose it turns out that a $2$-fair carpool system is possible but not a $1$-fair carpool system. Must $n$ be even?
[i]Proposed by Nathan Ramesh and Palmer Mebane
Three sides of a trapezoid are equal, and a circle with the longer base as a diameter halves the two non-parallel sides. Find the angles of the trapezoid.
Let $\triangle P_1P_2P_3$ be an equilateral triangle. For each $n\ge 4$, [i]Mingmingsan[/i] can set $P_n$ as the circumcenter or orthocenter of $\triangle P_{n-3}P_{n-2}P_{n-1}$. Find all positive integer $n$ such that [i]Mingmingsan[/i] has a strategy to make $P_n$ equals to the circumcenter of $\triangle P_1P_2P_3$.
[i]Proposed by Li4 and Untro368.[/i]
Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.
Let $g$ be the multiplicative function given by $$g(p^{\alpha}) = \alpha p^{\alpha-1},$$ for all $\alpha\in\mathbb Z^+$ and primes $p$. Prove that there exist infinitely many integers $n$ such that $$g(n+1) = g(n) + g(1).$$
A cockroach is crawling along the walls inside a cubical room that has an edge length of $\text{3 m}$. If the cockroach starts from the back lower left hand corner of the cube and finishes at the front upper right hand corner, what is the magnitude of the displacement of the cockroach?
(a) $\text{3}\sqrt{2} \ \text{m}$
(b) $\text{3}\sqrt[3]{2} \ \text{m}$
(c) $\text{3}\sqrt{3} \ \text{m}$
(d) $\text{3 m}$
(e) $\text{9 m}$
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
Problems 15, 16, and 17 all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.
16. Estimate the year in which the population of Nisos will be approximately 6,000.
$ \text{(A)}\ 2050\qquad\text{(B)}\ 2075\qquad\text{(C)}\ 2100\qquad\text{(D)}\ 2125\qquad\text{(E)}\ 2150 $
The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil
attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?
Let $n\ge 4$ be an even integer. A regular $n$-gon and a regular $(n-1)$-gon are inscribed into the unit circle. For each vertex of the $n$-gon consider the distance from this vertex to the nearest vertex of the $(n-1)$-gon, measured along the circumference. Let $S$ be the sum of these $n$ distances. Prove that $S$ depends only on $n$, and not on the relative position of the two polygons.
Find all real numbers $x$ such that: $$ \sqrt{\frac{x-7}{2015}}+\sqrt{\frac{x-6}{2016}}+\sqrt{\frac{x-5}{2017}}=\sqrt{\frac{x-2015}{7}}+\sqrt{\frac{x-2016}{6}}+\sqrt{\frac{x-2017}{5}}$$
Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$.[i](IMO Problem 1)[/i]
[b][i]Original formulation [/i][/b]
Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove:
(a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$
(b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n! $
[i]Proposed by Germany, FR[/i]
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.
Let $n$ points be given inside a rectangle $R$ such that no two of them lie on a line parallel to one of the sides of $R$. The rectangle $R$ is to be dissected into smaller rectangles with sides parallel to the sides of $R$ in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect $R$ into at least $n + 1$ smaller rectangles.
[i]Proposed by Serbia[/i]