Found problems: 85335
Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$?
$ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$
$\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$
A hollow box (with negligible thickness) shaped like a rectangular prism has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?
Tow circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.
Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2$, where $z_1$ lies on the segment with endpoints $3$ and $4i$, and $z_2$ has magnitude at most $1$. What integer is closest to the area of $\mathcal{R}$?
$\textbf{(A) }13\qquad\textbf{(B) }14\qquad\textbf{(C) }15\qquad\textbf{(D) }16\qquad\textbf{(E) }17$
Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.
Consider a right-angled triangle $ABC$, $\angle A = 90^{\circ}$ and points $D$ and $E$ on the leg $AB$ such that $\angle ACD \equiv \angle DCE \equiv \angle ECB$. Prove that if $3\overrightarrow{AD} = 2\overrightarrow{DE}$ and $\overrightarrow{CD} + \overrightarrow{CE} = 2\overrightarrow{CM}$ then $\overrightarrow{AB} = 4\overrightarrow{AM}$.
Let $a, b, c$ be nonnegative real numbers satisfying $a + b + c =3$.
a) If $c > \frac32$, prove that $3(ab + bc + ca) - 2abc < 7$.
b) Find the greatest possible value of $M =3(ab + bc + ca) - 2abc $.
A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:
$\textbf{(A) }\sqrt a:\sqrt b\qquad
\textbf{(B) }\sqrt a:\sqrt{a-b}\qquad
\textbf{(C) }\sqrt b:\sqrt{a-b}\qquad$
$\textbf{(D) }a:\sqrt{a-b}\qquad
\textbf{(E) }b:\sqrt{a-b}$
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$.
[i]Proposed by Deyuan Li and Andrew Milas[/i]
Let $\omega$ be a nonreal $47$th root of unity. Suppose that $\mathcal{S}$ is the set of polynomials of degree at most $46$ and coefficients equal to either $0$ or $1$. Let $N$ be the number of polynomials $Q \in \mathcal{S}$ such that
\[
\sum_{j = 0}^{46} \frac{Q(\omega^{2j}) - Q(\omega^{j})}{\omega^{4j} + \omega^{3j} + \omega^{2j} + \omega^j + 1} = 47.
\]
The prime factorization of $N$ is $p_1^{\alpha_1}p_2^{\alpha_2} \dots p_s^{\alpha_s}$ where $p_1, \ldots, p_s$ are distinct primes and $\alpha_1, \alpha_2, \ldots, \alpha_s$ are positive integers. Compute $\sum_{j = 1}^s p_j\alpha_j$.
Julia and Hansol are having a math-off. Currently, Julia has one more than twice as many points as Hansol. If Hansol scores $6$ more points in a row, he will tie Julia’s score. How many points does Julia have?
Let $ P(x)$ be a polynomial with a non-negative coefficients. Prove that if the inequality $ P\left(\frac {1}{x}\right)P(x)\geq 1$ holds for $ x \equal{} 1$, then this inequality holds for each positive $ x$.
Points $K$ and $L$ are on the sides $BC$ and $CD$, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.
A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time.
How tall is the building?
$\textbf{(A) } 50 \text{ m}\\
\textbf{(B) } 80 \text{ m}\\
\textbf{(C) } 100 \text{ m}\\
\textbf{(D) } 150 \text{ m}\\
\textbf{(E) } 400 \text{ m}$
For $a\in C^{*}$ find all $n\in N$ such that $X^{2}(X^{2}-aX+a^{2})^{2}$ divides $(X^{2}+a^{2})^{n}-X^{2n}-a^{2n}$
Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and
\[
a_k = (n-k+1) \cdot c_{k-1}, \quad
b_k = \binom nk - c_k - a_k, \quad \text{and} \quad
c_k = \frac{b_{k-1}}{k}
\]
for each integer $1 \leq k \leq n$.
$ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$.
Proposed by [i]Jonathan Du[/i]
a) Find the minimum positive integer $k$ so that for every positive integers $(x, y) $, for which $x/y^2$ and $y/x^2$, then $xy/(x+y) ^k$
b) Find the minimum positive integer $l$ so that for every positive integers $(x, y, z) $, for which $x/y^2$, $y/z^2$ and $z/x^2$, then $xyz/(x+y+z)^l$
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
[b]Q.[/b]Let $ABC$ be a triangle with incenter $I$ and orthocenter $H$. Let $A_1,A_2$ lie on $\overline{BC}$ such that $\overline{IA_1}\perp \overline{IB},\overline{IA_2}\perp\overline{IC}$. $\overline{AA_1},\overline{AA_2}$ cut $\odot(ABC)$ again at $A_3,A_4$. $\overline{A_3A_4}$ cuts $\overline{BC}$ at $A_0$. Similarly, we have $B_0,C_0$. Prove that $A_0,B_0,C_0$ are collinear on a line which is perpendicular to line $\overline{IH}$.
The equiangular convex hexagon $ ABCDEF$ has $ AB \equal{} 1$, $ BC \equal{} 4$, $ CD \equal{} 2$, and $ DE \equal{} 4$. The area of the hexagon is
$ \textbf{(A)}\ \frac{15}{2}\sqrt{3}\qquad
\textbf{(B)}\ 9\sqrt{3}\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ \frac{39}{4}\sqrt{3}\qquad
\textbf{(E)}\ \frac{43}{4}\sqrt{3}$
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$ .
Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.
A convex pentagon $ABCDE$ is given with $AB=BC$, $CD=DE$ and $\angle A=\angle C=\angle E>90^{\circ}$. Prove that the pentagon is circumscribed
[I]Proposed by F. Baharev[/i]
The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?
[asy]
size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4));
draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4));
[/asy]
$\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$
Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too.
[i]Proposed by Mirsaleh Bahavarnia, Iran[/i]
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.
[i]Proposed by Mohsen Jamali, Iran[/i]