Found problems: 85335
Given a graph with $99$ vertices and degrees in $\{81,82,\dots,90\}$, prove that there exist $10$ vertices of this graph with equal degrees and a common neighbour.
[i]Proposed by Alireza Alipour[/i]
The large square in the diagram below with sides of length $8$ is divided into $16$ congruent squares. Find the
area of the shaded region.
[img]https://cdn.artofproblemsolving.com/attachments/6/e/cf828197aa2585f5eab2320a43b80616072135.png[/img]
The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $1, 1\slash 2, 1\slash 3, \ldots.$ Each segment makes with the preceding a given angle $\theta.$ What is the distance and what is the direction of the limiting points (if there is one) from the initial point of the first segment?
Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that
a) triangle $AMC$ is isosceles.
b) circumcenter of triangle $AMC$ lies on circle $c$ .
[b]p1.[/b] What is the smallest possible value for the product of two real numbers that differ by ten?
[b]p2.[/b] Determine the number of positive integers $n$ with $1 \le n \le 400$ that satisfy the following:
$\bullet$ $n$ is a square number.
$\bullet$ $n$ is one more than a multiple of $5$.
$\bullet$ $n$ is even.
[b]p3.[/b] How many positive integers less than $2019$ are either a perfect cube or a perfect square but not both?
[b]p4.[/b] Felicia draws the heart-shaped figure $GOAT$ that is made of two semicircles of equal area and an equilateral triangle, as shown below. If $GO = 2$, what is the area of the figure?
[img]https://cdn.artofproblemsolving.com/attachments/3/c/388daa657351100f408ab3f1185f9ab32fcca5.png[/img]
[b]p5.[/b] For distinct digits $A, B$, and $ C$:
$$\begin{tabular}{cccc}
& A & A \\
& B & B \\
+ & C & C \\
\hline
A & B & C \\
\end{tabular}$$ Compute $A \cdot B \cdot C$.
[b]p6 [/b] What is the difference between the largest and smallest value for $lcm(a,b,c)$, where $a,b$, and $c$ are distinct positive integers between $1$ and $10$, inclusive?
[b]p7.[/b] Let $A$ and $B$ be points on the circumference of a circle with center $O$ such that $\angle AOB = 100^o$. If $X$ is the midpoint of minor arc $AB$ and $Y$ is on the circumference of the circle such that $XY\perp AO$, find the measure of $\angle OBY$ .
[b]p8. [/b]When Ben works at twice his normal rate and Sammy works at his normal rate, they can finish a project together in $6$ hours. When Ben works at his normal rate and Sammy works as three times his normal rate, they can finish the same project together in $4$ hours. How many hours does it take Ben and Sammy to finish that project if they each work together at their normal rates?
[b][b]p9.[/b][/b] How many positive integer divisors $n$ of $20000$ are there such that when $20000$ is divided by $n$, the quotient is divisible by a square number greater than $ 1$?
[b]p10.[/b] What’s the maximum number of Friday the $13$th’s that can occur in a year?
[b]p11.[/b] Let circle $\omega$ pass through points $B$ and $C$ of triangle $ABC$. Suppose $\omega$ intersects segment $AB$ at a point $D \ne B$ and intersects segment $AC$ at a point $E \ne C$. If $AD = DC = 12$, $DB = 3$, and $EC = 8$, determine the length of $EB$.
[b]p12.[/b] Let $a,b$ be integers that satisfy the equation $2a^2 - b^2 + ab = 18$. Find the ordered pair $(a,b)$.
[b]p13.[/b] Let $a,b,c$ be nonzero complex numbers such that $a -\frac{1}{b}= 8, b -\frac{1}{c}= 10, c -\frac{1}{a}= 12.$
Find $abc -\frac{1}{abc}$ .
[b]p14.[/b] Let $\vartriangle ABC$ be an equilateral triangle of side length $1$. Let $\omega_0$ be the incircle of $\vartriangle ABC$, and for $n > 0$, define the infinite progression of circles $\omega_n$ as follows:
$\bullet$ $\omega_n$ is tangent to $AB$ and $AC$ and externally tangent to $\omega_{n-1}$.
$\bullet$ The area of $\omega_n$ is strictly less than the area of $\omega_{n-1}$.
Determine the total area enclosed by all $\omega_i$ for $i \ge 0$.
[b]p15.[/b] Determine the remainder when $13^{2020} +11^{2020}$ is divided by $144$.
[b]p16.[/b] Let $x$ be a solution to $x +\frac{1}{x}= 1$. Compute $x^{2019} +\frac{1}{x^{2019}}$ .
[b]p17. [/b]The positive integers are colored black and white such that if $n$ is one color, then $2n$ is the other color. If all of the odd numbers are colored black, then how many numbers between $100$ and $200$ inclusive are colored white?
[b]p18.[/b] What is the expected number of rolls it will take to get all six values of a six-sided die face-up at least once?
[b]p19.[/b] Let $\vartriangle ABC$ have side lengths $AB = 19$, $BC = 2019$, and $AC = 2020$. Let $D,E$ be the feet of the angle bisectors drawn from $A$ and $B$, and let $X,Y$ to be the feet of the altitudes from $C$ to $AD$ and $C$ to $BE$, respectively. Determine the length of $XY$ .
[b]p20.[/b] Suppose I have $5$ unit cubes of cheese that I want to divide evenly amongst $3$ hungry mice. I can cut the cheese into smaller blocks, but cannot combine blocks into a bigger block. Over all possible choices of cuts in the cheese, what’s the largest possible volume of the smallest block of cheese?
PS. You had better use hide for answers.
Let $a$ and $b$ be prime numbers such that $a+b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.
Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.
Let $AA_1, BB_1$ be the altitudes of an acute non-isosceles triangle $ABC$. Circumference of the triangles $ABC$ meets that of the triangle $A_1B_1C$ at point $N$ (different from $C$). Let $M$ be the midpoint of $AB$ and $K$ be the intersection point of $CN$ and $AB$. Prove that the line of centers the circumferences of the triangles $ABC$ and $KMC$ is parallel to the line $AB$.
(I. Kachan)
The equation $ax^5 + bx^4 + c = 0$ has three distinct roots. Show that so does the equation $cx^5 +bx+a = 0$.
Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?
Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.
If the number $3n +1$, where n is integer, is multiple of $7$, find the possible remainders of the following divisions:
(a) of $n$ with divisor $7$,
(b) of $n^{m}$ with divisor $7$, for all values of the positive integer $m, m >1$.
Evaluate the infinite product $$\prod_{n=2}^{\infty} \frac{n^3-1}{n^3+1}.$$
Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.
Each girl among $100$ girls has $100$ balls; there are in total $10000$ balls in $100$ colors, from each color there are $100$ balls. On a move, two girls can exchange a ball (the first gives the second one of her balls, and vice versa). The operations can be made in such a way, that in the end, each girl has $100$ balls, colored in the $100$ distinct colors. Prove that there is a sequence of operations, in which each ball is exchanged no more than 1 time, and at the end, each girl has $100$ balls, colored in the $100$ colors.
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$ over the integers for every $i$.
$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that
$$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$
The circles $\mathcal C_1$ and $\mathcal C_2$ touch each other externally at $D$, and touch a circle $\omega$ internally at $B$ and $C$, respectively. Let $A$ be an intersection point of $\omega$ and the common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $D$. Lines $AB$ and $AC$ meet $\mathcal C_1$ and $\mathcal C_2$ again at $K$ and $L$, respectively, and the line $BC$ meets $\mathcal C_1$ again at $M$ and $\mathcal C_2$ again at $N$. Prove that the lines $AD$, $KM$, $LN$ are concurrent.
[i]Greece[/i]
A cake in the shape of a rectangular prism has dimensions $6\text{ cm}\times14\text{ cm}\times21\text{ cm}$. It is cut into $1764$ equally sized cubes such that each cube is $1\text{ cm}^3$. Andy the ant starts at one corner of the cake and eats through the cake in a straight line to the opposite corner of the cake. How many of the $1\text{ cm}^3$ cubes does Andy bite through?
Let $a, b, c ,d$ be real numbers greater than $1$ and $x, y$ be real numbers such that
\[ a^x+b^y = (a^2+b^2)^x \quad \text{and} \quad c^x+d^y = 2^y(cd)^{y/2} \]
Prove that $x<y$.
Triangles $ ABC$ and $ ADC$ are isosceles with $ AB \equal{} BC$ and $ AD \equal{} DC$. Point $ D$ is inside $ \triangle ABC$, $ \angle ABC \equal{} 40^\circ$, and $ \angle ADC \equal{} 140^\circ$. What is the degree measure of $ \angle BAD$?
$ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 60$
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.
Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$?
[i]By Omid Hatami[/i]
Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear.
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For any non-empty subset $X$ of $M=\{1,2,3,...,2021\}$, let $a_X$ be the sum of the greatest and smallest elements of $X$. Determine the arithmetic mean of all the values of $a_X$, as $X$ covers all the non-empty subsets of $M$.