Found problems: 85335
Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
Solve in positive integers $a>b$:
$$(a-b)^{ab}=a^bb^a$$
Consider a convex polyhedron whose faces are triangles. Prove that it is possible to color its edges by either red or blue, in a way that the following property is satisfied: one can travel from any vertex to any other vertex while passing only along red edges, and can also do this while passing only along blue edges.
Answer either (i) or (ii):
(i) A force acts on the element $ds$ of a closed plane curve. The magnitude of this force is $r^{-1} ds$ where $r$ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve, it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium.
(ii) Show that
$$x+ \frac{2}{3}x^{3}+ \frac{2\cdot 4}{3\cdot 5} x^5 +\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7} x^7 + \ldots= \frac{ \arcsin x}{\sqrt{1-x^{2}}}.$$
What is the median of the following list of $4040$ numbers$?$
$$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$
$\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$
Circles $k_1$ and $k_2$ with radii $r_1=6$ and $r_2=3$ are externally tangent and touch a circle $k$ with radius $r=9$ from inside. A common external tangent of $k_1$ and $k_2$ intersects $k$ at $P$ and $Q$. Determine the length of $PQ$.
Find all nonegative integers $ x,y,z$ such that $ 12^x\plus{}y^4\equal{}2008^z$
For every $ x,y\in \Re ^{\plus{}}$, the function $ f: \Re ^{\plus{}} \to \Re$ satisfies the condition $ f\left(x\right)\plus{}f\left(y\right)\equal{}f\left(x\right)f\left(y\right)\plus{}1\minus{}\frac{1}{xy}$. If $ f\left(2\right)<1$, then $ f\left(3\right)$ will be
$\textbf{(A)}\ 2/3 \\ \textbf{(B)}\ 4/3 \\ \textbf{(C)}\ 1 \\ \textbf{(D)}\ \text{More information needed} \\ \textbf{(E)}\ \text{There is no } f \text{ satisfying the condition above.}$
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
$ \text{(A)}\ 150 \qquad
\text{(B)}\ 160 \qquad
\text{(C)}\ 170 \qquad
\text{(D)}\ 180 \qquad
\text{(E)}\ 190$
* Prove that if the trihedral angles at each of the vertices of a triangular pyramid are formed by the identical planar angles, then all faces of this pyramid are equal.
Say an integer polynomial is $\textit{primitive}$ if the greatest common divisor of its coefficients is $1$. For example, $2x^2+3x+6$ is primitive because $\gcd(2,3,6)=1$. Let $f(x)=a_2x^2+a_1x+a_0$ and $g(x) = b_2x^2+b_1x+b_0$, with $a_i,b_i\in\{1,2,3,4,5\}$ for $i=0,1,2$. If $N$ is the number of pairs of polynomials $(f(x),g(x))$ such that $h(x) = f(x)g(x)$ is primitive, find the last three digits of $N$.
The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.
$5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?
We define [i]similar numbers[/i] as positive integers that have exactly the same digits (but possibly in another order). For example, $1241$, $2114$ and $4211$ are similar numbers, but $1424$ is not similar to the other three.
Determine whether there exist three similar numbers, each with $300$ digits (all digits being non-zero), such that the sum of two of them equals the third. If the answer is yes, provide an example; if not, justify why it is impossible.
$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \in \{1, 2, \dots, 1001\} = S$. $B$ must guess the value of $n$ by choosing several subsets of $S$, then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecting $k_1, k_2$ then $k_3$ subsets of $S$ each.
What is the least value of $k_1 + k_2 + k_3$ such that $B$ has a strategy to correctly guess the value of $n$ no matter what $A$ chooses?
let $x_1$, $x_2$ .... $x_6$ be non-negative reals such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5$ + $x_2x_4x_6$ $\geq$ $\frac{1}{540}$. Let $p$ and $q$ be relatively prime integers such that $\frac{p}{q}$ is the maximum value of $x_1x_2x_3+x_2x_3x_4+x_3x_4x_5+x_4x_5x_6+x_5x_6x_1+x_6x_1x_2$. Find $p+q$
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$ be the eight vertices of a $30 \times30\times30$ cube as shown. The two figures $ACFH$ and $BDEG$ are congruent regular tetrahedra. Find the volume of the intersection of these two tetrahedra.
[asy]
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draw((5,1)--(7,1), linewidth(1.6));
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draw((7,1)--(8,1.5));
draw((5,1)--(6,1.5));
draw((6,3.5)--(6,1.5));
draw((6,1.5)--(8,1.5));
draw((8,3.5)--(8,1.5), linewidth(1.6));
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label("$ F $",(6.93,0.96),SE*labelscalefactor);
label("$ G $",(8.07,1.64),SE*labelscalefactor);
label("$ H $",(5.65,1.90),SE*labelscalefactor);
dot((-3,3),dotstyle);
dot((-3,1),dotstyle);
dot((-1,3),dotstyle);
dot((-1,1),dotstyle);
dot((-2,3.5),dotstyle);
dot((0,3.5),dotstyle);
dot((0,1.5),dotstyle);
dot((-2,1.5),dotstyle);
dot((1,3),dotstyle);
dot((1,1),dotstyle);
dot((3,3),dotstyle);
dot((3,1),dotstyle);
dot((2,3.5),dotstyle);
dot((4,3.5),dotstyle);
dot((4,1.5),dotstyle);
dot((2,1.5),dotstyle);
dot((5,3),dotstyle);
dot((5,1),dotstyle);
dot((6,3.5),dotstyle);
dot((7,3),dotstyle);
dot((7,1),dotstyle);
dot((8,3.5),dotstyle);
dot((8,1.5),dotstyle);
dot((6,1.5),dotstyle); [/asy]
Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?
At a two-round volleyball tournament participated 99 teams. Each played a match at home and a match away. Each team won exactly half of their home matches and exactly half of their away matches. Prove that one of the teams beat another team twice.
[i]Proposed by M. Antipov[/i]
On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.
Let $E$ be the point of intersection of the diagonals of convex quadrilateral $ABCD$, and let $P,Q,R,$ and $S$ be the centers of the circles circumscribing triangles $ABE,$ $BCE$, $CDE$, and $ADE$, respectively. Then
$\textbf{(A) }PQRS\text{ is a parallelogram}$
$\textbf{(B) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rhombus}$
$\textbf{(C) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a rectangle}$
$\textbf{(D) }PQRS\text{ is a parallelogram if an only if }ABCD\text{ is a parallelogram}$
$\textbf{(E) }\text{none of the above are true}$
Two triangles have equal longest sides and equal smallest angles. A new triangle is constructed, such that its sides are the sum of the longest sides, the sum of the shortest sides, and the sum of the middle sides of the initial triangles. Prove that the area of the new triangle is at least twice as much as the sum of the areas of the initial ones.
Find the sum of the positive integer solutions to the equation $\left\lfloor\sqrt[3]{x}\right\rfloor+\left\lfloor\sqrt[4]{x}\right\rfloor=4.$