Found problems: 85335
Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had 43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
A)$46$ B)$50$ C)$48$ D)$47$ E)$49$
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.
Let $n$ be a natural number with more than $2021$ digits, none of which are $8$ or $9$. Suppose that $n$ has no common factors with $2021$. Prove that it is possible to increase one of the digits of $n$ by at most $2$ so that the resulting number is a multiple of 2021.
A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be written as
\[ \frac{2\left(\frac{m\pi}{n}+\frac{\sqrt{p}}{q}\right)^2}{\pi^2} \]
for positive integers $m$, $n$, $p$, and $q$ such that $m$ and $n$ are relatively prime and $p$ is squarefree. Find $m+n+p+q$.
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.
we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.
In a box of $100$ marbles, just $3\%$ of the marbles are purple and the rest are green. How many green marbles must be removed from the box so that $95\%$ of the remaining marbles are green?
$\text{(A) }2\qquad\text{(B) }15\qquad\text{(C) }37\qquad\text{(D) }40\qquad\text{(E) }57$
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation
\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]
holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
A criminal is at point $X$, and three policemen at points $A, B$ and $C$ block him up, i.e. the point $X$ lies inside the triangle $ABC$. Each evening one of the policemen is replaced in the following way: a new policeman takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points $A, B$ and $C$ (it is known that at any moment $X$ does not lie on a side of the triangle)?
by V.Protasov
Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that
$\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$.
[i]Proposed by Mirko Petrusevski[/i]
Determine, if they exist, the real values of $x$ and $y$ that satisfy that $$\frac{x^2}{y^2} +\frac{y^2}{x^2} +\frac{x}{y}+\frac{y}{x} = 0$$ such that $x + y <0.$
Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ($1\le i\le n,1\le j\le k$) give pairwise different residues modulo $nk$?
Cris has the equation $-2x^2 + bx + c = 0$, and Cristian increases the coefficients of the Cris equation by $1$, obtaining the equation $-x^2 + (b + 1) x + (c + 1) = 0$. Mariloli notices that the real solutions of the Cristian's equation are the squares of the real solutions of the Cris equation. Find all possible values that can take the coefficients $b$ and $c$.
Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.
In a certain triangle, the bisectors of the two interior angles were extended to the intersection with the circumscribed circle and two equal chords were obtained. Is it true that the triangle is isosceles?
Let \(ABCD\) be a convex quadrilateral in the plane and let \(O_{A}, O_{B}, O_{C}\) and \(O_{D}\) be the circumcenters of the triangles \(BCD, CDA, DAB\) and \(ABC\), respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.
Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.
The pair of positive integers $(a, b)$ is such that a does not divide $b$, $b$ does not divide a, both numbers are at most $100$, and they have the maximal possible number of common divisors. What is the largest possible value of $a \cdot· b$?
A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$.
(Karl Czakler)
Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $
[b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term.
[b]b)[/b] Is $ 2015 $ the largest element of one of these sets?
Two congruent rectangles are located as shown in the figure. Find the area of the shaded part.
[img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]
For given integer $n \geq 3$, set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$. Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$, one of these number does not lie in between the other two numbers in every permutations $p_i$ ($1 \leq i \leq m$). (For example, in the permutation $(1, 3, 2, 4)$, $3$ lies in between $1$ and $4$, and $4$ does not lie in between $1$ and $2$.) Determine the maximum value of $m$.
Prove that the number $\tan \frac{\pi}{3^n}$ is irrational for any natural $n$.
An ellipse $\Gamma$ and its chord $AB$ are given. Find the locus of orthocenters of triangles $ABC$ inscribed into $\Gamma$.