Found problems: 85335
Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$. Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Show that $A^{2}+3 B^{2}= 4$.
Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.
One hundred points labeled $1$ to $100$ are arranged in a $10\times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels $1$ to $10,$ the second row has labels $11$ to $20,$ and so on).
Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by $7$ is either on the boundary or in the interior of $\mathcal{P}.$ Compute the smallest possible area of $\mathcal{P}.$
In the attached figure, point $C$ is the center of the circle, $AB$ is tangent to the circle, $P-C-P'$ and $AC\perp PP'$. If $AT = 2$ cm. and $AB = 4$ cm, calculate $BQ$
[img]https://cdn.artofproblemsolving.com/attachments/e/e/d47429b82fb87299c40f5224489313909cfd0f.png[/img]
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
Find the least positive integer $k$ such that when $\frac{k}{2023}$ is written in simplest form, the sum of the numerator and denominator is divisible by $7$.
[i]Proposed byMuztaba Syed[/i]
Let $ABC$ be an acute triangle, and let $D,E,F$ be the feet of its altitudes from $A,B,C$ respectively. The lines $AB{}$ and $DE$ cross at $K{}$ and the lines $AC$ and $DF$ cross at $L{}.$ Let $M$ be the midpoint of the side $BC$ and let the line $AM$ cross the circle $(ABC)$ again at $N{}.$ The parallel through $M{}$ to $EF$ crosses the line $KL$ at $P{}.$ Prove that the triangle $MNP$ is isosceles.
Call the intersection of two segments [i]almost perfect[/i] if for each of the segments the distance between the midpoint of the segment and the intersection is at least $2022$ times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is [i]almost perfect[/i].
[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.
Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \leq N \leq 2^{2002}$. Prove that it is possible to color every subset of $S$ either black or white so that the following conditions hold:
(a) the union of any two white subsets is white;
(b) the union of any two black subsets is black;
(c) there are exactly $N$ white subsets.
Let $a_1,a_2,\ldots,a_7, b_1,b_2,\ldots,b_7\geq 0$ be real numbers satisfying $a_i+b_i\le 2$ for all $i=\overline{1,7}$.
Prove that there exist $k\ne m$ such that $|a_k-a_m|+|b_k-b_m|\le 1$.
Thanks for show me the mistake typing
For $n$ people, if it is known that
(a) there exist two people knowing each other among any three people, and
(b) there exist two people not knowing each other among any four people.
Find the maximum of $n$.
Here, we assume that if $A$ knows $B$, then $B$ knows $A$.
A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Ayase randomly picks a number $x \in (0, 1]$ with uniform probability. He then draws the six points $(0, 0, 0),(x, 0, 0),(2x, 3x, 0),(5, 5, 2),(7, 3, 0),(9, 1, 4)$. If the expected value of the volume of the convex polyhedron formed by these six points can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m + n$
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that
\[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]
Let $n>1$ be an odd integer, and let $a_1$, $a_2$, $\dots$, $a_n$ be distinct real numbers. Let $M$ be the maximum of these numbers and $m$ the minimum. Show that it is possible to choose the signs of the expression $s=\pm a_1\pm a_2\pm\dots\pm a_n$ so that
\[m<s<M\]
Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
$\textbf{(A) }\dfrac{\sqrt3}4\qquad
\textbf{(B) }\dfrac{\sqrt3}3\qquad
\textbf{(C) }\dfrac23\qquad
\textbf{(D) }\dfrac{\sqrt2}2\qquad
\textbf{(E) }\dfrac{\sqrt3}2$
The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that
a) there are three consecutive numbers with the sum being at least $15$,
b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.
A triangular table with $n$ rows and $n$ columns is given, the $i$-th row ends with a field in the $v$-th column. In each field of the table, some of the numbers $1,2,..., n$ are written such that for each $k \in {1, 2,..., n}$ all the numbers $1,2,..., n$ occur in the union of the $k$-th row and the $k$-th column.
Prove that for odd $n$, each of the numbers $1,2,..., n$ is written in the last box of a row.
[img]https://cdn.artofproblemsolving.com/attachments/f/9/2aed55628edb1505c7de27c152127b04d8d991.png[/img]
How many integers $x$, from $10$ to $99$ inclusive, have the property that the remainder of $x^2$ divided by $100$ is equal to the square of the units digit of $x$?
For integers $0 \le m,n \le 2^{2017}-1$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $2^{2017} \times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 2^{2017}$) is \[ (-1)^{\alpha(i-1, j-1)}. \] For $1 \le i, j \le 2^{2017}$, let $M_{i,j}$ be the matrix with the same entries as $M$ except for the $(i,j)$th entry, denoted by $a_{i,j}$, and such that $\det M_{i,j}=0$. Suppose that $A$ is the $2^{2017} \times 2^{2017}$ matrix whose $(i,j)$th entry is $a_{i,j}$ for all $1 \le i, j \le 2^{2017}$. Compute the remainder when $\det A$ is divided by $2017$.
[i]Proposed by Michael Ren and Ashwin Sah[/i]
For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions.
Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$.
[i]David Yang.[/i]
Find $x$ such that \[2010^{\log_{10}x}=11^{\log_{10}(1+3+5+\cdots +4019).}\]
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.