Found problems: 85335
A circle has radius $52$ and center $O$. Points $A$ is on the circle, and point $P$ on $\overline{OA}$ satisfies $OP = 28$. Point $Q$ is constructed such that $QA = QP = 15$, and point $B$ is constructed on the circle so that $Q$ is on $\overline{OB}$. Find $QB$.
[i]Proposed by Justin Hsieh[/i]
Let $A$ and $B$ be arbitrary finite sets and let $f: A\longrightarrow B$ and $g: B\longrightarrow A$
be functions such that $g$ is not onto. Prove that there is a subset $S$ of $A$ such that
$\frac{A}{S}=g(\frac{B}{f(S)})$.
$n\geq 4$ is an integer number. For any permutation $x_1,x_2,\cdots,x_n$ of the numbers $1,2 \cdots,n$ we write the number
$$
x_1+2x_2+\cdots+nx_n
$$
on the board. Compute the number of total distinct numbers written on the board.
Characterize the set of positive integers $n$ such that, for all integers $a$, the sequence $a$, $a^2$, $a^3$, $\cdots$ is periodic modulo $n$.
$P$, $Q$, $R$ and $S$ are (distinct) points on a circle. $PS$ is a diameter and $QR$ is parallel to the diameter $PS$. $PR$ and $QS$ meet at $A$. Let $O$ be the centre of the circle and let $B$ be chosen so that the quadrilateral $POAB$ is a parallelogram. Prove that $BQ$ = $BP$ .
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).
Suppose that $x,y,z\in [0,1]$. Find the maximal value of the expression
\[\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}.\]
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y,z$, we have $$(f(x)+1)(f(y)+f(z))=f(xy+z)+f(xz-y)$$
Let $ABCD$ be a convex quadrilateral with $A,B,C,D$ concyclic. Assume $\angle ADC$ is acute and $\frac{AB}{BC}=\frac{DA}{CD}$. Let $\Gamma$ be a circle through $A$ and $D$, tangent to $AB$, and let $E$ be a point on $\Gamma$ and inside $ABCD$.
Prove that $AE\perp EC$ if and only if $\frac{AE}{AB}-\frac{ED}{AD}=1$.
Gizmo is thinking of a geometric sequence in which the third term is $1215$ and the fifth is $540$. Which of the following could be the eighth term of Gizmo's sequence?
$\text{(A) }-160\qquad\text{(B) }-135.5\qquad\text{(C) }216\qquad\text{(D) }240\qquad\text{(E) }472.5$
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$
7. What is the minimum value of the product $$\prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}}$$ given that $(a_1, a_2, a_3, a_4, a_5, a_6)$ is a permutation of $(1, 2, 3, 4, 5, 6)$? (note $a_7 = a_1, a_8 = a_2 \ldots$)
8. Danielle picks a positive integer $1 \le n \le 2016$ uniformly at random. What is the probability that $\text{gcd}(n, 2015) = 1$?
9. How many $3$-element subsets of the set $\{1, 2, 3, . . . , 19\}$ have sum of elements divisible by $4$?
$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule:
1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students;
2) Each student received the maximum possible points in each problem or got $ 0$ in it;
Lasha got the least number of points. What's the maximal number of points he could have?
Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 \minus{} k$ points in it.
Given any positive integer $c$, denote $p(c)$ as the largest prime factor of $c$. A sequence $\{a_n\}$ of positive integers satisfies $a_1>1$ and $a_{n+1}=a_n+p(a_n)$ for all $n\ge 1$. Prove that there must exist at least one perfect square in sequence $\{a_n\}$.
There are a number of arcs on the edge of a circular disk. Each pair of arcs has the least one point in common. Show that on the circle you can choose two diametrical opposites points such that each arc contains at least one of these two points.
Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map.
[b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing.
[b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $
[b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $
[i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
[i]Proposed by Mariusz Skałba, Poland[/i]
Find the maximum number $ c$ such that for all $n \in \mathbb{N}$ to have \[ \{n \cdot \sqrt{2}\} \geq \frac{c}{n}\] where $ \{n \cdot \sqrt{2}\} \equal{} n \cdot \sqrt{2} \minus{} [n \cdot \sqrt{2}]$ and $ [x]$ is the integer part of $ x.$ Determine for this number $ c,$ all $ n \in \mathbb{N}$ for which $ \{n \cdot \sqrt{2}\} \equal{} \frac{c}{n}.$
A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity).
(a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$.
(b) Show that, for $n$ even, no such tiling is possible
If $m,~n,~p,$ and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution
$\textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad$
$\textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad$
$\textbf{(C) }\text{if and only if }mq-np=0\qquad$
$\textbf{(D) }\text{if and only if }n(1-p)-q(1-m)=0\qquad$
$\textbf{(E) }\text{if and only if }(1-n)(1-p)-(1-q)(1-m)=0$
It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n) = 10n, g(n) = 10n + 4$, and $h(2n) = n$; in other words, one may write $0$ or $4$ in the end of the number and one may divide an even number by $2$. Prove: every positive integer can be constructed starting from $4$ and performing a finite number of the operations $f,
g,$ and $h$ in some order.
Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$
Let $a,b,c$ be distinct complex numbers with $|a|=|b|=|c|=1.$ If $|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12,$ prove that the points of affixes $a,b,c$ are the vertices of an equilateral triangle.
A convex polyhedron is floating in a sea. Can it happen that $90\%$ of its volume is below the water level, while more than half of its surface area is above the water level?
(A Shapovalov)