Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $ 20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $ 10$ vertical feet above the bottom? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7.5 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 15$

Let $P$ be a point inside a triangle $ABC$, $d_a$, $d_b$ and $d_c$ be distances from $P$ to the lines $BC$, $AC$ and $AB$ respectively, $R$ be a radius of the circumcircle and $r$ be a radius of the inscribed circle for $\Delta ABC.$ Prove that: $$\sqrt{d_a}+\sqrt{d_b}+\sqrt{d_c}\leq\sqrt{2R+5r}.$$

Let $ABCD$ be a rectangle of center $O$ in the plane $\alpha$, and let $V \notin\alpha$ be a point in space such that $V O \perp \alpha$. Let $A' \in (V A)$, $B'\in (V B)$, $C'\in (V C)$, $D'\in (V D)$ be four points, and let $M$ and $N$ be the midpoints of the segments $A'C'$ and $B'D'$. .Prove that $MN \parallel \alpha$ if and only if $V , A', B', C', D'$ all lie on a sphere.

By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)

Let $x$ and $y$ be two positive reals. Prove that $xy\le\frac{x^{n+2}+y^{n+2}}{x^n+y^n}$ for all non-negative integers $n$.

Given a triangle $ABC$. We construct two angle bisectors in the corners $A$ and $B$. Than we construct two lines parallel to those ones through the point $C$. $D$ and $E$ are intersections of those lines with the bisectors. It happens, that $(DE)$ line is parallel to $(AB)$. Prove that the triangle is isosceles.

Let $C$ be an interior point of line segment $AB$. Equilateral triangles $ADC$ and $CEB$ are constructed to the same side from $AB$. Find all points which can be the midpoint of the segment $DE$.

Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n = \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.

Let $x, y$ be positive real numbers. If \[129-x^2=195-y^2=xy,\] then $x = \frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m+n$. [i]Proposed by Michael Tang

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that: a) ${F_{p + 1}} \equiv 0(\bmod p).$ b) $k(p)|2p+2.$ c) $k(p)$ is divisible by $4.$

Let there be positive integers $a, c$. Positive integer $b$ is a divisor of $ac-1$. For a positive rational number $r$ which is less than $1$, define the set $A(r)$ as follows. $$A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \}$$ Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$.

Let $n$ be a positive integer . If $n$ has odd number of divisors ( other than $1$ and $n$ ) , then show that $n$ is a perfect square .

A surjective function $g: \mathbb{C} \to \mathbb C$ is given. Find all functions $f: \mathbb{C} \to \mathbb C$ such that for all $x,y\in \mathbb C$ we have $$ |f(x)+g(y)| = | f(y) + g(x)|. $$ Proposed by [i]Mojtaba Zare, Amirabbas Mohammadi[/i]

Let $ABC$ be an obtuse triangle with obtuse angle $\angle B$ and altitudes $AD, BE, CF$. Let $T$ and $S$ be the midpoints of $AD$ and $CF$, respectively. Let $M$ and $N$ and be the symmetric images of $T$ with respect to lines $BE$ and $BD$, respectively. Prove that $S$ lies on the circumcircle of triangle $BMN$.

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.

Let equilateral triangle $\vartriangle ABC$ be inscribed in a circle $\omega_1$ with radius $4$. Consider another circle $\omega_2$ with radius $2$ internally tangent to $\omega_1$ at $A$. Let $\omega_2$ intersect sides $\overline{AB}$ and $\overline{AC}$ at $D$ and $E$, respectively, as shown in the diagram. Compute the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/a/5/8255cdb8b041719d735607da8139aa4016375d.png[/img]

Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that [list=disc] [*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and [*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$. [/list] Prove that $\max(a_1,a_{2023})\ge 507$.

Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i=1}^4 x_i=98.$ Find $\frac n{100}.$

Let $ U \equal{} 2\times2004^{2005}$, $ V \equal{} 2004^{2005}$, $ W \equal{} 2003\times2004^{2004}$, $ X \equal{} 2\times2004^{2004}$, $ Y \equal{} 2004^{2004}$ and $ Z \equal{} 2004^{2003}$. Which of the following is the largest? $ \textbf{(A)}\ U \minus{} V \qquad \textbf{(B)}\ V \minus{} W \qquad \textbf{(C)}\ W \minus{} X \qquad \textbf{(D)}\ X \minus{} Y \qquad \textbf{(E)}\ Y \minus{} Z$

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

A man buys a house for $10000$ dollars and rents it. He puts $ 12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325$ dollars a year taxes and realizes $ 5\frac{1}{2}\%$ on his investment. The monthly rent is: $\textbf{(A)}\ 64.82\text{ dollars} \qquad \textbf{(B)}\ 83.33\text{ dollars} \qquad \textbf{(C)}\ 72.08\text{ dollars} \qquad \textbf{(D)}\ 45.83\text{ dollars} \qquad \textbf{(E)}\ 177.08\text{ dollars}$

[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x,y,z$ be positive real numbers satisfying $4x^2 - 2xy + y^2 = 64, y^2 - 3yz +3z^2 = 36,$ and $4x^2 +3z^2 = 49.$ If the maximum possible value of $2xy +yz -4zx$ can be expressed as $\sqrt{n}$ for some positive integer $n,$ find $n.$

On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k\geq 0$ lines. How many different values of $k$ are possible? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$