Found problems: 85335
In a small city there are $n$ bus routes, with $n > 1$, and each route has exactly $4$ stops. If any two routes have exactly one common stop, and each pair of stops belongs to exactly one route, find all possible values of $n$.
Find all triples $(x,y,z)$ of real numbers such that
\[x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).\]
A census man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said "We do not mind giving you the sum of the ages of any two ladies you may choose". Thereupon, the census man said, "In that case, please give me the sum of the ages of every possible pair of you". They gave the sums as: 30, 33, 41, 58, 66, 69. The census man took these figures and happily went away.
How did he calculate the individual ages?
$AB$ and $CD$ are segments lying on the two sides of an angle whose vertex is $O$. $A$ is between $O$ and $B$, and $C$ is between $O$ and $D$ . The line connecting the midpoints of the segments $AD$ and $BC$ intersects $AB$ at $M$ and $CD$ at $N$. Prove that $\frac{OM}{ON}=\frac{AB}{CD}$
(V Senderov)
Each of the $50$ students in a class sent greeting cards to $25$ of the others. Prove that there exist two students who greeted each other.
Let $a,b,c$, and $d$ be real numbers such that
$$a+b = c +d+ 12$$
and
$$ab + cd - 28 = bc + ad.$$
Find the minimum possible value of $a^4+b^4+c^4+d^4$.
28. The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least $2$ larger than an adjacent number.
29. We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) if an American $A_1$ attacks another American $A_2$, then $A_2$ also attacks $A_1$. Let $m$ be the number of squares that an American attacks when placed in the top left corner of an 8 by 8 chessboard. Let $n$ be the maximal number of Americans that can be placed on the $8$ by $8$ chessboard such that no Americans attack each other, if one American must be in the top left corner. Find the largest possible value of $mn$.
30. On the blackboard, Amy writes $2017$ in base-$a$ to get $133201_a$. Betsy notices she can erase a digit from Amy’s number and change the base to base-$b$ such that the value of the the number remains the same. Catherine then notices she can erase a digit from Betsy’s number and change the base to base-$c$ such that the value still remains the same. Compute, in decimal, $a + b + c$.
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$
where $i = 1, 2$.
Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.
Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$.
[i]Linus Tang[/i]
A triangle $T$ has all integer side lengths and at most one of its side lengths is greater than ten. What is the largest possible area of $T$ ?
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
Edward is trying to spell the word "CAT". He has an equal chance of spelling the word in any order of letters (i.e. TAC or TCA). What is the probability that he spells "CAT" incorrectly?
$\textbf{(A) }\dfrac16\qquad\textbf{(B) }\dfrac13\qquad\textbf{(C) }\dfrac12\qquad\textbf{(D) }\dfrac23\qquad\textbf{(E) }\dfrac56$
A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
Let $ABC$ be an isosceles triangle with right angle at point $A$. Find the minimum of the function $F$ given by
\[F(M) = BM +CM-\sqrt{3}AM\]
In the following diagram, the square and pentagon are both regular and share segment $AB$ as designated. What is the measure of $\angle CBD$ in degrees?
Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$
Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.
Given that $x$ and $y$ are real numbers, compute the minimum value of
$$x^4+4x^3+8x^2+4xy+6x+4y^2+10.$$
Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with center $O$. Let $F$ be the second intersection point of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the intersection point of the segments $AC$ and $BD$.
(A Zaslavskiy)
Given a triangle $ABC$. On its sides $BC$ , $CA$ and $AB$ , the points $A_1$ , $B_1$ and $C_1$ are taken, respectively , such that
$2 \angle B_1 A_1 C_1 + \angle BAC = 180^o$ ,
$2 \angle A_1 C_1 B_1 + \angle ACB = 180^o$ ,
$2 \angle C_1 B_1 A_1 + \angle CBA = 180^o$ .
Find the locus of the centers of the circles circumscribed about the triangles $A_1 B_1 C_1$ (all possible such triangles are considered).
Let $a,b$ and $c$ be integers and $r$ a real number such that $ar^2+br+c=0$ with $ac\not =0$.Prove that $\sqrt{r^2+c^2}$ is an irrational number
The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.
Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number.
The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is:
$\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$
An automobile went up a hill at a speed of $ 10$ miles an hour and down the same distance at a speed of $ 20$ miles an hour. The average speed for the round trip was:
$ \textbf{(A)}\ 12\frac {1}{2} \text{ mph} \qquad\textbf{(B)}\ 13\frac {1}{3} \text{ mph} \qquad\textbf{(C)}\ 14\frac {1}{2} \text{ mph} \qquad\textbf{(D)}\ 15 \text{ mph}$
$ \textbf{(E)}\ \text{none of these}$
Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$.
Proposed by Mahdi Etesamifard and Alireza Dadgarnia