A car has a 4-digit integer price, which is written digitally. (so in digital numbers, like on your watch probably) While the salesmen isn't watching, the buyer turns the price upside down and gets the car for 1626 less. How much did the car initially cost?

We know that $201$ and $9$ give the same remainder when divided by $24$. What is the smallest positive integer $k$ such that $201+k$ and $9+k$ give the same remainder when divided by $24+k$? [i]2020 CCA Math Bonanza Lightning Round #2.1[/i]

Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.

Calculate the sum of vector products $[[x, y], z] + [[y, z], x] + [[z, x], y]$

For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$. (For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, \(A\), \(B\), \(C\), which can each be inscribed in a circle with radius \(1\). Let \(\varphi_A\) denote the measure of the acute angle made by the diagonals of quadrilateral \(A\), and define \(\varphi_B\) and \(\varphi_C\) similarly. Suppose that \(\sin\varphi_A=\frac{2}{3}\), \(\sin\varphi_B=\frac{3}{5}\), and \(\sin\varphi_C=\frac{6}{7}\). All three quadrilaterals have the same area \(K\), which can be written in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

Define the operation $ \star$ by $ a\star b \equal{} (a \plus{} b)b$. What is $ (3\star 5) \minus{} (5\star 3)$? $ \textbf{(A)}\ \minus{}16\qquad \textbf{(B)}\ \minus{}8\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$ A. Voidelevich

Find all functions $f\colon\mathbb R^+\to\mathbb R^+$ such that \[(f(x)+f(y)+f(z))(xf(y)+yf(z)+zf(x))>(f(x)+y)(f(y)+z)(f(z)+x)\] for all $x,y,z\in\mathbb R^+$. [i]Alexander Wang[/i] [size=59](oops)[/size]

Let $\{a_n\}$ be the sequence such that $a_0=2019$ and $$a_n=-\frac{2020}{n}\sum_{k=0}^{n-1}a_k.$$ Compute the last three digits of $\sum_{n=1}^{2020}2020^na_nn$.

A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12% $ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?

A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?

In a class of $26$ students, everyone is being graded on five subjects with one of three possible marks. Prove that if $25$ of these students have received their marks, then we can grade the last one in such a way that their marks differ from these of any other student on at least two subjects.

Suppose that $f : \mathbb{R}^+\to\mathbb R$ satisfies the equation $$f(a+b+c+d) = f(a)+f(b)+f(c)+f(d)$$ for all $a,b,c,d$ that are the four sides of some tangential quadrilateral. Show that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}^+$.

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 $\max(a_1,a_{2023})\ge 507$.

[u]Set 7[/u] [b]G19.[/b] How many ordered triples $(x, y, z)$ with $1 \le x, y, z \le 50$ are there such that both $x + y + z$ and $xy + yz + zx$ are divisible by$ 6$? [b]G20.[/b] Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. If $D$ is the foot of the perpendicular from $A$ to $BC$, then $AH = 8$ and $HD = 3$. If $\angle AOH = 90^o$, find $BC^2$. [b]G21.[/b] Nate flips a fair coin until he gets two heads in a row, immediately followed by a tails. The probability that he flips the coin exactly $12$ times is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 8[/u] [b]G22.[/b] Let $f$ be a function defined by $f(1) = 1$ and $$f(n) = \frac{1}{p}f\left(\frac{n}{p}\right)f(p) + 2p - 2,$$ where $p$ is the least prime dividing $n$, for all integers $n \ge 2$. Find $f(2022)$. [b]G23.[/b] Jessica has $15$ balls numbered $1$ through $15$. With her left hand, she scoops up $2$ of the balls. With her right hand, she scoops up $2$ of the remaining balls. The probability that the sum of the balls in her left hand is equal to the sum of the balls in her right hand can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]G24.[/b] Let $ABCD$ be a cyclic quadrilateral such that its diagonal $BD = 17$ is the diameter of its circumcircle. Given $AB = 8$, $BC = CD$, and that a line $\ell$ through A intersects the incircle of $ABD$ at two points $P$ and $Q$, the maximum area of $CP Q$ can be expressed as a fraction $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$. [u]Set 9[/u] [i]This set consists of three estimation problems, with scoring schemes described.[/i] [b]G25.[/b] Estimate $N$, the total number of participants (in person and online) at MOAA this year. An estimate of $e$ gets a total of max $ \left( 0, \lfloor 150 \left( 1- \frac{|N-e|}{N}\right) \rfloor -120 \right)$ points. [b]G26.[/b] If $A$ is the the total number of in person participants at MOAA this year, and $B$ is the total number of online participants at MOAA this year, estimate $N$, the product $AB$. An estimate of $e$ gets a total of max $(0, 30 - \lceil \log10(8|N - e| + 1)\rceil )$ points. [b]G27.[/b] Estimate $N$, the total number of letters in all the teams that signed up for MOAA this year, both in person and online. An estimate of e gets a total of max $(0, 30 - \lceil 7 log5(|N - E|)\rceil )$ points. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 4-6 [url=https://artofproblemsolving.com/community/c3h3131305p28367080]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Let $A\% B = BA - B - A + 1$. How many digits are in the number $1\%(3\%(3\%7))$ ? [b]p2. [/b]Three circles, of radii $1, 2$, and $3$ are all externally tangent to each other. A fourth circle is drawn which passes through the centers of those three circles. What is the radius of this larger circle? [b]p3.[/b] Express $\frac13$ in base $2$ as a binary number. (Which, similar to how demical numbers have a decimal point, has a “binary point”.) [b]p4. [/b] Isosceles trapezoid $ABCD$ with $AB$ parallel to $CD$ is constructed such that $DB = DC$. If $AD = 20$, $AB = 14$, and $P$ is the point on $AD$ such that $BP + CP$ is minimized, what is $AP/DP$? [b]p5.[/b] Let $f(x) = \frac{5x-6}{x-2}$ . Define an infinite sequence of numbers $a_0, a_1, a_2,....$ such that $a_{i+1} = f(a_i)$ and $a_i$ is always an integer. What are all the possible values for $a_{2014}$ ? [b]p6.[/b] $MATH$ and $TEAM$ are two parallelograms. If the lengths of $MH$ and $AE$ are $13$ and $15$, and distance from $AM$ to $T$ is $12$, find the perimeter of $AMHE$. [b]p7.[/b] How many integers less than $1000$ are there such that $n^n + n$ is divisible by $5$ ? [b]p8.[/b] $10$ coins with probabilities of $1, 1/2, 1/3 ,..., 1/10$ of coming up heads are flipped. What is the probability that an odd number of them come up heads? [b]p9.[/b] An infinite number of coins with probabilities of $1/4, 1/9, 1/16, ...$ of coming up heads are all flipped. What is the probability that exactly $ 1$ of them comes up heads? [b]p10.[/b] Quadrilateral $ABCD$ has side lengths $AB = 10$, $BC = 11$, and $CD = 13$. Circles $O_1$ and $O_2$ are inscribed in triangles $ABD$ and $BDC$. If they are both tangent to $BD$ at the same point $E$, what is the length of $DA$ ? PS. You had better use hide for answers.

$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.

Prove that the transformation product of the symmetry of center $(0, 0)$ with the symmetry of the axis, with the line of equation $x = y + 1$, can be expressed as a product of an axis symmetry the line $e$ by a translation of vector $\overrightarrow{v}$, with $e$ parallel to $\overrightarrow{v}$, . Determine a line $e$ and a vector $\overrightarrow{v}$, that meet the indicated conditions. have to be unique $e$ and $\overrightarrow{v}$,?

Three lines are drawn parallel to the sides of the triangles in the opposite to the vertex, not belonging to the side, part of the plane. The distance from each side to the corresponding line equals the length of the side. Prove that six intersection points of those lines with the continuations of the sides are situated on one circumference.

\[\int_0^1 \frac{1}{x+\sqrt x}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]

Fix a convex $n > 3$ gon $A_{1}A_{2}...A_{n} $ and connect every two points with a road. Call this $n$-gon [i]crossy[/i] if no three roads intersect at a point inside the polygon. This $n$-gon is partitioned into a set $S$ of disjoint polygons formed by the roads. Label every intersection with an integer such that $A_{1}$ is non-zero. Call the labelling [i]basic[/i] if for every polygon in $S$, the sum of the labels of its vertices is $0$. $(a)$ Prove that there is a [i]basic[/i] labelling of a crossy $n$-gon when $n$ is even. $(b)$ Prove that there is no [i]basic[/i] labelling of a crossy $n$-gon when $n$ is odd.

Suppose $N$ is an $n$ digit positive integer such that (a) all its digits are distinct; (b) the sum of any three consecutive digits is divisible by $5$. Prove that $n \leq 6$. Further, show that starting with any digit, one can find a six digit number with these properties.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x + y) + y \le f(f(f(x)))\] holds for all $x, y \in \mathbb{R}$.