Annie and Bonnie are running laps around a 400-meter oval track. They started together, but Annie has pulled ahead because she is $25 \%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie? $\textbf{(A) }1 \frac{1}{4}\qquad\textbf{(B) }3 \frac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25$

Let $n = 6901$. There are $6732$ positive integers less than or equal to $n$ that are also relatively prime to $n$. Find the sum of the distinct prime factors of $n$.

Let $p$ and $q$ be prime numbers with $p<q$. Suppose that in a convex polygon $P_1,P_2,…,P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that $$P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}$$holds for every integer $k$ with $1\leqslant k\leqslant p$. [i]Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin[/i]

Shannon, Laura, and Tasha found a shirt which came in five colors at their favorite store, and they each bought one of each color of that shirt. On Monday, they all wear one of their new shirts to work. What is the probability that Shannon, Laura, and Tasha will not all be wearing the same color shirt that day? $\text{(A) }\frac{12}{25}\qquad\text{(B) }\frac{16}{25}\qquad\text{(C) }\frac{21}{25}\qquad\text{(D) }\frac{22}{25}\qquad\text{(E) }\frac{24}{25}$

Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$. [i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$, where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.

Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively. [i]Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov[/i]

If $ a\ge b>1$, what is the largest possible value of $ \log_a(a/b)\plus{}\log_b(b/a)$? $ \textbf{(A)}\ \minus{}2 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Given any odd integer $n>3$ that is not divisible by $3$, determine whether it is possible to fill an $n \times n$ grid with $n^2$ integers such that (each cell filled with a number, the number at the intersection of the $i$-th row and $j$-th column is denoted as $a_{ij}$): $\cdot$ Each row and each column contains a permutation of the numbers $0,1,2, \cdots, n-1$. $\cdot$ The pairs $(a_{ij},a_{ji})$ for $i<j$ are all distinct.

Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$: $$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$ [i]Proposed by Sina Saleh[/i]

What is the area of the region defined by the inequality $ |3x\minus{}18|\plus{}|2y\plus{}7|\le 3$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{9}{2} \qquad \textbf{(E)}\ 5$

Let $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$ and let $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$. Find the value of the following expression: $$f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4).$$

What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month?

The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.

There are positive real numbers $a$, $b$, $c$, $d$, and $p$ such that $a$ is $62.5\%$ of $b$, $b$ is $64\%$ of $c$, c is $125\%$ of $d$, and $d$ is $p\%$ of $a$. Find $p$.

Given positive odd number $m$ and integer ${a}.$ Proof: For any real number $c,$ $$\#\left\{x\in\mathbb Z\cap [c,c+\sqrt m]\mid x^2\equiv a\pmod m\right\}\le 2+\log_2m.$$ [i]Proposed by Yinghua Ai[/i]

For every positive integer $n{}$ denote $a_n$ as the last digit of the sum of the number from $1$ to $n{}$. For example $a_5=5, a_6=1.$ a) Find $a_{21}.$ b) Compute the sum $a_1+a_2+\ldots+a_{2015}.$

Color the vertexes of a quadrangular pyramid in one color, satisfying that two end points of any edge are in different colors. We have only 5 colors, then the number of ways coloring the quadrangular pyramid is________.

A permutation $\pi : \{1,2,\ldots,N\} \rightarrow \{1,2, \ldots,N\}$ is [i]very odd[/i] if the smallest positive integer $k$ such that $\pi^k(a) = a$ for all $1 \le a \le N$ is odd, where $\pi^k$ denotes $\pi$ composed with itself $k$ times. Let $X_0 = 1,$ and for $i \ge 1,$ let $X_i$ be the fraction of all permutations of $\{1,2,\ldots,i\}$ that are very odd. Let $S$ denote the set of all ordered $4$-tuples $(A,B,C,D)$ of nonnegative integers such that $A+B +C +D =2023.$ Find the last three digits of the integer $$2023\sum_{(A,B,C,D) \in S} X_AX_BX_CX_D.$$

When Troy writes his digits, his $0$, $1$, and $8$ look the same right-side-up and upside-down as seen in the figure below. His $6$ and $9$ look like upside-down images of each other. None of his other digits look like digits when they are inverted. How many different five-digit numbers (which do not begin with the digit zero) can Troy write which read the same right-side-up and upside-down? [asy] frame l; label(l,"\textsf{0}\qquad \textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}"); add(rotate(180)*l); label("\textsf{0}\qquad\textsf{l}\qquad\textsf{2}\qquad\textsf{3}\qquad\textsf{4}\qquad\textsf{5}\qquad\textsf{6}\qquad\textsf{7}\qquad\textsf{8}\qquad\textsf{9}",(0,20)); [/asy]

Let $O$ be the circumcenter and $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $T$ is the midpoint of the segment $AO$. The perpendicular bisector of $AO$ intersects the line $BC$ at point $S$. Prove that the circumcircle of the triangle $AST$ bisects the segment $OH$. [i](M. Berindeanu, RMC 2018 book)[/i]

A frog trainer places one frog at each vertex of an equilateral triangle $ABC$ of unit sidelength. The trainer can make one frog jump over another along the line joining the two, so that the total length of the jump is an even multiple of the distance between the two frogs just before the jump. Let $M$ and $N$ be two points on the rays $AB$ and $AC$, respectively, emanating from $A$, such that $AM = AN = \ell$, where $\ell$ is a positive integer. After a finite number of jumps, the three frogs all lie in the triangle $AMN$ (inside or on the boundary), and no more jumps are performed. Determine the number of final positions the three frogs may reach in the triangle $AMN$. (During the process, the frogs may leave the triangle $AMN$, only their nal positions are to be in that triangle.)

Find the maximum number of elements which can be chosen from the set $ \{1,2,3,\ldots,2003\}$ such that the sum of any two chosen elements is not divisible by 3.

Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

Let $\mathcal S$ be the sphere with center $(0,0,1)$ and radius $1$ in $\mathbb R^3$. A plane $\mathcal P$ is tangent to $\mathcal S$ at the point $(x_0,y_0,z_0)$, where $x_0$, $y_0$, and $z_0$ are all positive. Suppose the intersection of plane $\mathcal P$ with the $xy$-plane is the line with equation $2x+y=10$ in $xy$-space. What is $z_0$?

[b]p1.[/b] Compute $$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$ [b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elements in each subset is divisible by $3$? [b]p3.[/b] Let $a^2$ and $b^2$ be two integers. Consider the triangle with one vertex at the origin, and the other two at the intersections of the circle $x^2 + y^2 = a^2 + b^2$ with the graph $ay = b|x|$. If the area of the triangle is numerically equal to the radius of the circle, what is this area? [b]p4.[/b] Suppose $f(x) = x^3 + x - 1$ has roots $a$, $b$ and $c$. What is $$\frac{a^3}{1-a}+\frac{b^3}{1-b}+\frac{c^3}{1-c} ?$$ [b]p5.[/b] Lisa has a $2D$ rectangular box that is $48$ units long and $126$ units wide. She shines a laser beam into the box through one of the corners such that the beam is at a $45^o$ angle with respect to the sides of the box. Whenever the laser beam hits a side of the box, it is reflected perfectly, again at a $45^o$ angle. Compute the distance the laser beam travels until it hits one of the four corners of the box. [b]p6.[/b] How many ways can we form a group with an odd number of members (plural) from $99$ people total? Express your answer in the form $a^b + c$, where $a$, $b$, and $c$ are integers, and $a$ is prime. [b]p7.[/b] Let $$S = \log_2 9 \log_3 16 \log_4 25 ...\log_{999} 1000000.$$ Compute the greatest integer less than or equal to $\log_2 S$. [b]p8.[/b] A prison, housing exactly four hundred prisoners in four hundred cells numbered $1$-$400$, has a really messed-up warden. One night, when all the prisoners are asleep and all of their doors are locked, the warden toggles the locks on all of their doors (that is, if the door is locked, he unlocks the door, and if the door is unlocked, he locks it again), starting at door $1$ and ending at door $400$. The warden then toggles the lock on every other door starting at door $2$ ($2$, $4$, $6$, etc). After he has toggled the lock on every other door, the warden then toggles every third door (doors $3$, $6$, $9$, etc.), then every fourth door, etc., finishing by toggling every $400$th door (consisting of only the $400$th door). He then collapses in exhaustion. Compute the number of prisoners who go free (that is, the number of unlocked doors) when they wake up the next morning. [b]p9.[/b] Let $A$ and $B$ be fixed points on a $2$-dimensional plane with distance $AB = 1$. An ant walks on a straight line from point $A$ to some point $C$ on the same plane and finds that the distance from itself to $B$ always decreases at any time during this walk. Compute the area of the locus of points where point $C$ could possibly be located. [b]p10.[/b] A robot starts in the bottom left corner of a $4 \times 4$ grid of squares. How many ways can it travel to each square exactly once and then return to its start if it is only allowed to move to an adjacent (not diagonal) square at each step? [b]p11.[/b] Assuming real values for $p$, $q$, $r$, and $s$, the equation $$x^4 + px^3 + qx^2 + rx + s$$ has four non-real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Find $q$. [b]p12.[/b] A cube is inscribed in a right circular cone such that one face of the cube lies on the base of the cone. If the ratio of the height of the cone to the radius of the cone is $2 : 1$, what fraction of the cone's volume does the cube take up? Express your answer in simplest radical form. [b]p13.[/b] Let $$y =\dfrac{1}{1 +\dfrac{1}{9 +\dfrac{1}{5 +\dfrac{1}{9 +\dfrac{1}{5 +...}}}}}$$ If $y$ can be represented as $\frac{a\sqrt{b} + c}{d}$, where $b$ is not divisible by the square of any prime, and the greatest common divisor of $a$ and $d$ is $1$, find the sum $a + b + c + d$. [b]p14.[/b] Alice wants to paint each face of an octahedron either red or blue. She can paint any number of faces a particular color, including zero. Compute the number of ways in which she can do this. Two ways of painting the octahedron are considered the same if you can rotate the octahedron to get from one to the other. [b]p15.[/b] Find $n$ in the equation $$133^5 + 110^5 + 84^5 + 27^5 = n^5,$$ where $n$ is an integer less than $170$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].