Found problems: 85335
Let $a$ and $b$ be positive real numbers. Consider a regular hexagon of side $a$, and build externally on its sides six rectangles of sides $a$ and $b$. The new twelve vertices lie on a circle. Now repeat the same construction, but this time exchanging the roles of $a$ and $b$; namely; we start with a regular hexagon of side $b$ and we build externally on its sides six rectangles of sides $a$ and $b$. The new twelve vertices lie on another circle.
Show that the two circles have the same radius.
Let a real number $\lambda > 1$ be given and a sequence $(n_k)$ of positive integers such that $\frac{n_{k+1}}{n_k}> \lambda$ for $k = 1, 2,\ldots$ Prove that there exists a positive integer $c$ such that no positive integer $n$ can be represented in more than $c$ ways in the form $n = n_k + n_j$ or $n = n_r - n_s$.
Let $n$ be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular $2n$-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let $P(n)$ denote the number of different ways Luca can make $2n$ jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.)
[b]a)[/b] Find the value of $P(n)$ if $n$ is odd.
[b]b)[/b] Prove that if $n$ is even, then
\[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\]
[i]Proposed by Péter Csikvári and Kartal Nagy, Budapest[/i]
In each square of a $4\times 4$ board, we are to write an integer in such a way that the sums of the numbers in each column and in each row are nonnegative integral powers of $2$. Is it possible to do this in such a way that every two of these eight sums are different? Justify your answer.
In triangle $ABC$ we have $AB = BC, \angle B = 20^o$. Point $M$ on $AC$ is such that $AM : MC = 1 : 2$, point $H$ is the projection of $C$ to $BM$. Find angle $AHB$.
(M. Yevdokimov)
Let $n > 1$ be an odd natural number. The squares of an $n \times n$ chessboard are alternately colored white and black so that the four corner squares are black. An $L$-triomino is an $L$-shaped piece that covers exactly three squares of the board. For which values of $n$ is it possible to cover all black squares with $L$-triominoes, so that no two $L$-triominos overlap? For these values of $n$ determine the smallest possible number of $L$-triominoes that are necessary for this.
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$.
\[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\]
If necessary, you may use $ \ln 3 \equal{} 1.10$.
Show that the triangle whose angles satisfy the equality \[ \frac{sin^2(A) + sin^2(B) + sin^2(C)}{cos^2(A) + cos^2(B) + cos^2(C)} = 2 \] is a rectangular triangle.
Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$. Consider sequences $x_1,\dots , x_n$ with the following properties:
(i) $k$ terms $x_i$, including $x_1$, are equal to $a$;
(ii) $m$ terms $x_i$, including $x_n$, are equal to $b$;
(iii) no three consecutive terms are equal.
Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$.
Given a regular pentagon of area $1$, a pivot line is a line not passing through any of the pentagon's vertices such that there are $3$ vertices of the pentagon on one side of the line and $2$ on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points.
We call a natural number $p{}$ [i]simple[/i] if for any natural number $k{}$ such that $2\leqslant k\leqslant \sqrt{p}$ the inequality $\{p/k\}\geqslant 0,01$ holds. Is the set of simple prime numbers finite?
[i]Proposed by M. Didin[/i]
Which of the following methods of proving a geometric figure a locus is not correct?
$ \textbf{(A)}\ \text{Every point of the locus satisfies the conditions and every point not on the locus does not satisfy the conditions.}$
$ \textbf{(B)}\ \text{Every point not satisfying the conditions is not on the locus and every point on the locus does satisfy the conditions.}$
$ \textbf{(C)}\ \text{Every point satisfying the conditions is on the locus and every point on the locus satisfies the conditions.}$
$ \textbf{(D)}\ \text{Every point not on the locus does not satisfy the conditions and every point not satisfying} \\
\text{the conditions is not on the locus.}$
$ \textbf{(E)}\ \text{Every point satisfying the conditions is on the locus and every point not satisfying the conditions is not on the locus.}$
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers: \begin{align*}700\cdot 712\,\cdot\, &718+320,\\701\cdot 713\,\cdot\, &719+320,\\ 702\cdot 714\,\cdot\, &720+320,\\&\vdots\\798\cdot 810\,\cdot\, &816+320,\\799\cdot 811\,\cdot\, &817+320.\end{align*} She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.
$A, B, C, D, E$ are five points on the same circle, so that $ABCDE$ is convex and we have $AB = BC$ and $CD = DE$. Suppose that the lines $(AD)$ and $(BE)$ intersect at $P$, and that the line $(BD)$ meets line $(CA)$ at $Q$ and line $(CE)$ at $T$. Prove that the triangle $PQT$ is isosceles.
(I. Voronovich)
It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$
$N $ different numbers are written on blackboard and one of these numbers is equal to $0$.One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between $-2016$ and $2016$ were written on blackboard(and some other numbers maybe). Find the smallest possible value of $N $.
Let $S$ be a non-empty set of positive integers such that for any $n \in S$, all positive divisors of $2^n+1$ are also in $S$. Prove that $S$ contains an integer of the form $(p_1p_2 \ldots p_{2023})^{2023}$, where $p_1, p_2, \ldots, p_{2023}$ are distinct prime numbers, all greater than $2023$.
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
In a concyclic quadrilateral $PQRS$,$\angle PSR=\frac{\pi}{2}$ , $H,K$ are perpendicular foot from $Q$ to sides $PR,RS$ , prove that $HK$ bisect segment$SQ$.
[b]p1.[/b] Find the number of ordered triples $(a, b, c)$ satisfying
$\bullet$ $a, b, c$ are are single-digit positive integers, and
$\bullet$ $a \cdot b + c = a + b \cdot c$.
[b]p2.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.)
[b]p3.[/b] Suppose that $a + \frac{1}{b} = 2$ and $b +\frac{1}{a} = 3$. If$ \frac{a}{b} + \frac{b}{a}$ can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$.
[b]p4.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1$$
Find the sum of all possible values of $10A + B$.
[b]p5.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $AC$ and $BC$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE = EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a, b$ positive integers and a squarefree, then find $a + b$.
[b]p6.[/b] Five bowling pins $P_1$, $P_2$,..., $P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i -j| = 1$.)
[b]p7.[/b] Let triangle $ABC$ have side lengths $AB = 10$, $BC = 24$, and $AC = 26$. Let $I$ be the incenter of $ABC$. If the maximum possible distance between $I$ and a point on the circumcircle of $ABC$ can be expressed as $a +\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.
(The incenter of any triangle $XY Z$ is the intersection of the angle bisectors of $\angle Y XZ$, $\angle XZY$, and $\angle ZY X$.)
[b]p8.[/b] How many terms in the expansion of
$$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$
have coefficients equal to $1011$?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $n \ge 2$ be a fixed positive integer. An integer will be called "$n$-free" if it is not a multiple of an $n$-th power of a prime. Let $M$ be an infi nite set of rational numbers, such that the product of every $n$ elements of $M$ is an $n$-free integer. Prove that $M$ contains only integers.
An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle.
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light?
[b]Note:[/b] 1 foot is equal to 12 inches.
$\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 18.5 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 20.5 \qquad\textbf{(E)}\ 22.5 $
Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\,
b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$
Prove that $$|a-b| <\frac{1}{99! 100!}$$
(G Galperin, Moscow)
Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.