Found problems: 85335
A mathematics competition had $9$ easy and $6$ difficult problems. Each of the participants in the competition solved $14$ of the $15$ problems. For each pair, consisting of an easy and a difficult problem, the number of participants who solved both those problems was recorded. The sum of these recorded numbers was $459$. How many participants were there?
A game is played on a $1 \times 1000$ board. There are n chips, all of which are initially in a box near the board. Two players move in turn. The first may choose $17$ chips or less, from either on or off the board. She then puts them into unoccupied cells on the board so that there is no more than one chip in each of the cells. The second player may take off the board any number of chips occupying consecutive cells and put them back in the box. The first player wins if she can put all n chips on the board so that they occupy consecutive cells.
(a) Show that she can win if $n = 98$.
(b) For what maximal value of $n$ can she win?
(A Shapovalov)
Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are equal?
[i]Proposed by Belarus for the 1999th IMO[/i]
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.
[u]Calculus Round[/u]
[b]p1.[/b] Evaluate $$\int^2_{-2}(x^3 + 2x + 1)dx$$
[b]p2.[/b] Find $$\lim_{x \to 0} \frac{ln(1 + x + x^3) - x}{x^2}$$
[b]p3.[/b] Find the largest possible value for the slope of a tangent line to the curve $f(x) = \frac{1}{3+x^2}$ .
[b]p4.[/b] An empty, inverted circular cone has a radius of $5$ meters and a height of $20$ meters. At time $t = 0$ seconds, the cone is empty, and at time $t \ge 0$ we fill the cone with water at a rate of $4t^2$ cubic meters per second. Compute the rate of change of the height of water with respect to time, at the point when the water reaches a height of $10$ meters.
[b]p5.[/b] Compute $$\int^{\frac{\pi}{2}}_0 \sin (2016x) \cos (2015x) dx$$
[b]p6.[/b] Let $f(x)$ be a function defined for $x > 1$ such that $f''(x) = \frac{x}{\sqrt{x^2-1}}$ and $f'(2) =\sqrt3$. Compute the length of the graph of $f(x)$ on the domain $x \in (1, 2]$.
[b]p7.[/b] Let the function $f : [1, \infty) \to R$ be defuned as $f(x) = x^{2 ln(x)}$. Compute $$\int^{\sqrt{e}}_1 (f(x) + f^{-1}(x))dx$$
[b]p8.[/b] Calculate $f(3)$, given that $f(x) = x^3 + f'(-1)x^2 + f''(1)x + f'(-1)f(-1)$.
[b]p9.[/b] Compute $$\int^e_1 \frac{ln (x)}{(1 + ln (x))^2} dx$$
[b]p10.[/b] For $x \ge 0$, let $R$ be the region in the plane bounded by the graphs of the line $\ell$ : $y = 4x$ and $y = x^3$. Let $V$ be the volume of the solid formed by revolving $R$ about line $\ell$. Then $V$ can be expressed in the form $\frac{\pi \cdot 2^a}{b\sqrt{c}}$ , where $a$, $b$, and $c$ are positive integers, $b$ is odd, and $c$ is not divisible by the square of a prime. Compute $a + b + c$.
[u]Calculus Tiebreaker[/u]
[b]Tie 1.[/b] Let $f(x) = x + x(\log x)^2$. Find $x$ such that $xf'(x) = 2f(x)$.
[b]Tie 2.[/b] Compute $$\int^{\frac{\sqrt2}{2}}_{-1} \sqrt{1 - x^2} dx$$
[b]Tie 3.[/b] An axis-aligned rectangle has vertices at $(0,0)$ and $(2, 2016)$. Let $f(x, y)$ be the maximum possible area of a circle with center at $(x, y)$ contained entirely within the rectangle. Compute the expected value of $f$ over the rectangle.
PS. You should use hide for answers.
A chess piece is placed on one of the squares of an $8\times 8$ chessboard where it begins a tour of the board: it moves from square to square, only moving horizontally or vertically. It visits every square precisely once, and ends up exactly where it started. What is the maximum number of times the piece can change direction along its tour?
[list]
[*]There are two semi-infinite plane mirrors inclined physically at a non-zero angle with the inner surfaces being reflective.\\
Prove that all lines of incident/reflected rays are tangential to a particular circle for any given incident ray being incident on a reflective side. Assume that the incident ray lies on one of the normal planes to the mirrors.[/*]
[*] There's a cone of an arbitrary base with large enough length.\\
The inner surface polished (Outer surface is absorbing in nature) and the apex is fixed to a point. The cone is being rotated around the apex at an angular speed $\omega$ around the vertical axis and assume that a large part of the inside is visible horizontally. A fixed horizontal ray is projected from outside towards the cone (which often falls inside of it), prove that all the lines of incident ray/reflected rays at all instants lie tangential to a particular sphere.\\
Try guessing the radius of the sphere with the parameters you observe.[/*]
Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$.
a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$;
b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.
The ratio $ \frac{2^{2001}\cdot3^{2003}}{6^{2002}}$ is
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{2}{3} \qquad
\textbf{(E)}\ \frac{3}{2}$
The fraction $\frac{3}{10}$ can be written as a sum of two reciprocals in exactly two ways:
\[\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}\]
a) In how many ways can $\frac{3}{2021}$ be written as a sum of two reciprocals?
b) Is there a positive integer $n$ not divisible by $3$ with the property that $\frac{3}{n}$ can be written as a sum of two reciprocals in exactly $2021$ ways?
A piece of paper is folded in half. A second fold is made at an angle $ \phi$ ($ 0^\circ < \phi < 90^\circ$) to the first, and a cut is made as shown below.
[img]12881[/img]
When the piece of paper is unfolded, the resulting hole is a polygon. Let $ O$ be one of its vertices. Suppose that all the other vertices of the hole lie on a circle centered at $ O$, and also that $ \angle XOY \equal{} 144^\circ$, where $ X$ and $ Y$ are the the vertices of the hole adjacent to $ O$. Find the value(s) of $ \phi$ (in degrees).
Compute the maximum value attained by $f(x)=x^{1/x^2}$.
Find all ordered pairs $(x,y)$ of real numbers that satisfy the following system of equations:
$$\begin{cases}
y(x+y)^2=2\\
8y(x^3-y^3) = 13.
\end{cases}$$
Given the function $$y =|x^2 - 4x + 3|.$$
Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.
In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality:
\[2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.\]
Given three points $A, B$ and $C$ (not collinear) construct the equilateral triangle of greater perimeter such that each of its sides passes through one of the given points.
Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties:
1) $1 \le ai \le p$ for all $i = 1, . . . , p$.
2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$.
3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.
Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$.
Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
Let $x$, $y$, and $z$ be real numbers (not necessarily positive) such that $x^4+y^4+z^4+xyz=4$.
Show that $x\le2$ and $\sqrt{2-x}\ge\frac{y+z}{2}$.
[i]Proposed by Alyazeed Basyoni[/i]
For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$
[i]Proposed by Suhaimi Ramly, Malaysia[/i]
$ABCD$ is a convex quadrilateral. Take the $12$ points which are the feet of the altitudes in the triangles $ABC, BCD, CDA, DAB$. Show that at least one of these points must lie on the sides of $ABCD$.
You have a collection of $\$20.21$, consisting of pennies, nickels, and quarters. To reduce the collection’s worth to $k$ cents, you simultaneously replace all pennies with quarters and all quarters with pennies (all coins are replaced one time). What is the minimum possible $k$?
$\textbf{(A) }105\qquad\textbf{(B) }120\qquad\textbf{(C) }125\qquad\textbf{(D) }505\qquad\textbf{(E) }101$
Let $ABC$ be an acute triangle with $\angle ACB>2 \angle ABC$. Let $I$ be the incenter of $ABC$, $K$ is the reflection of $I$ in line $BC$. Let line $BA$ and $KC$ intersect at $D$. The line through $B$ parallel to $CI$ intersects the minor arc $BC$ on the circumcircle of $ABC$ at $E(E \neq B)$. The line through $A$ parallel to $BC$ intersects the line $BE$ at $F$.
Prove that if $BF=CE$, then $FK=AD$.