Found problems: 85335
In triangle $ABC$ with $AB = 26$, $BC = 28$, and $C A = 30$, let $M$ be the midpoint of $AB$ and let $N$ be the midpoint of $C A$. The circumcircle of triangle $BCM$ intersects $AC$ at $X\ne C$, and the circumcircle of triangle $BCN $intersects $AB$ at $Y\ne B$. Lines $MX$ and $NY$ intersect $BC$ at $P$ and $Q$, respectively. The area of quadrilateral $PQY X$ can be expressed as $\frac{p}{q}$ for positive integers $p$ and $q$ such that gcd$(p,q) = 1$. Find $q$.
What is the greatest possible value of c such that $x^2+5x+c=0$ has at least one real solution?
Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.
The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.
How many positive integers $n$ are there such that there are $20$ positive integers that are less than $n$ and relatively prime with $n$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{None}$
Show that for any real numbers $a_{3},a_{4},...,a_{85}$, not all the roots of the equation $a_{85}x^{85}+a_{84}x^{84}+...+a_{3}x^{3}+3x^{2}+2x+1=0$ are real.
Do either (1) or (2):
(1) Prove that the determinant of the matrix
$$\begin{pmatrix}
1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\
2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\
2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2
\end{pmatrix}$$
is given by $(1+a^2 +b^2 +c^2)^{3}$.
(2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.
Consider the intersection of an ellipsoid with a plane $ \sigma$ passing through its center $ O$. On the line through the point $ O$ perpendicular to $ \sigma$, mark the two points at a distance from $ O$ equal to the area of the intersection. Determine the loci of the marked points as $ \sigma$ runs through all such planes.
[i]L. Tamassy[/i]
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$.
In the cyclic quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at $P$. Let $O$ be the center of the circumcircle $ABCD$, and $E$ a point of the extension of $OC$ beyond $C$. A parallel line to $CD$ is drawn through $E$ that cuts the extension of $OD$ beyonf $D$ at $F$. Let $Q$ be a point interior to $ABCD$, such that $\angle AFQ = \angle BEQ$ and $\angle FAQ = \angle EBQ$. Prove that $PQ \perp CD$.
Let $a_1\geq a_2\geq \cdots \geq a_n >0 .$ Prove that$$
\left( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)^2\geq \sum_{k=1}^{n} \frac{k(2k-1)}{a^2_1+a^2_2+\cdots+a^2_k}$$
Prove this proposition: Center the sphere circumscribed around a tetrahedron which coincides with the center of a sphere inscribed in that tetrahedron if and only if the skew edges of the tetrahedron are equal.
Let $ f(x)\equal{}x^2\plus{}6x\plus{}1$, and let $ R$ denote the set of points $ (x,y)$ in the coordinate plane such that
\[ f(x)\plus{}f(y)\le0\text{ and }f(x)\minus{}f(y)\le0
\]The area of $ R$ is closest to
$ \textbf{(A)}\ 21 \qquad
\textbf{(B)}\ 22 \qquad
\textbf{(C)}\ 23 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 25$
Let $A{}$ be a subset formed of $16$ elements of the set $B=\{1, 2, 3, \ldots, 105, 106\}$ such that the difference between every two elements from $A$ is different from $6, 9, 12, 15, 18, 21$. Prove that there are two elements in $A{}$ whose difference is $3$.
As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
Beto has rectangular chessboard where the number of rows and columns are consecutive numbers (for example, $30$ and $31$). Ana has tiles of two colors and different sizes: the red tiles are $5 \times 7$ rectangles and the blue tiles are $3 \times 5$ rectangles. Ana realized that she can cover all the squares of Beto’s board using only red tiles, which can be rotated, but must not overlap or extend beyond the board. Then, she realized she can also do the same using only blue tiles. What is the minimum number of squares that Beto’s board can have?
Let $a,b$ be real numbers such that $|a| \neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$ Find the value of the expression $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$
Find the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \le x, y \le 2166$.
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]
Given two circle $M$ and $N$, we say that $M$ bisects $N$ if they intersect in two points and the common chord is a diameter of $N$. Consider two fixed non-concentric circles $C_1$ and $C_2$.
a) Show that there exists infinitely many circles $B$ such that $B$ bisects both $C_1$ and $C_2$.
b) Find the locus of the centres of such circles $B$.
Points $P_1, P_2, P_3,$ and $P_4$ are $(0,0), (10, 20), (5, 15),$ and $(12, -6)$, respectively. For what point $P \in \mathbb{R}^2$ is the sum of the distances from $P$ to the other $4$ points minimal?
We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation:
\[*x^4+*x^3+*x^2+*x^1+*=0\]
with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy
Point $P$ lies inside square $ABCD$ such that the areas of $\triangle{PAB}, \triangle{PBC}, \triangle{PCD},$ and $\triangle{PDA}$ are $1, 2, 3,$ and $4,$ in some order. Compute $PA \cdot PB \cdot PC \cdot PD.$
Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy:
$$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\
(x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\
...\\
(x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$
Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.