Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

Each day, Jenny ate $ 20\%$ of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, $ 32$ remained. How many jellybeans were in the jar originally? $ \textbf{(A)}\ 40\qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60\qquad \textbf{(E)}\ 75$

In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$.

Let $n$ be a non-zero natural number, $n \ge 5$. Consider $n$ distinct points in the plane, each colored or white, or black. For each natural $k$ , a move of type $k, 1 \le k <\frac {n} {2}$, means selecting exactly $k$ points and changing their color. Determine the values of $n$ for which, whatever $k$ and regardless of the initial coloring, there is a finite sequence of $k$ type moves, at the end of which all points have the same color.

In triangle $ABC$, points $M$, $N$ are the midpoints of sides $AB$, $AC$ respelctively. Let $D$ and $E$ be two points on line segment $BN$ such that $CD \parallel ME$ and $BD <BE$. Prove that $BD=2\cdot EN$.

Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$.

The faces of a convex polyhedron are similar triangles. Prove that this polyhedron has two pairs of congruent faces.

The square $ABCD$ is divided into $n^2$ equal small (elementary) squares by parallel lines to its sides, (see the figure for the case $n = 4$). A spider starts from point$ A$ and moving only to the right and up tries to arrive at point $C$. Every ” movement” of the spider consists of: ”$k$ steps to the right and $m$ steps up” or ”$m$ steps to the right and $k$ steps up” (which can be performed in any way). The spider first makes $l$ ”movements” and in then, moves to the right or up without any restriction. If $n = m \cdot l$, find all possible ways the spider can approach the point $C$, where $n, m, k, l$ are positive integers with $k < m$. [img]https://cdn.artofproblemsolving.com/attachments/2/d/4fb71086beb844ca7c492a30c7d333fa08d381.png[/img]

Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$.

Let $\ell_1$, $\ell_2$, $\ell_3$m $\ell_4$ be rays from the origin, intersecting the line $y = 1$ at $x$-coordinates $-8$,$-2$, $1$, $2$, respectively. For $-\frac18 < k < \frac12$ , the line $y = 1 + kx$ intersects the four rays at points $A,B,C,D$, respectively. When $\overline{AB} = \overline{CD}$, the ratio $\frac{\overline{AB}}{\overline{BC}}$ is $\frac{p}{q}$ where $p, q$ are relatively prime positive integers. Find $p + q$.

How many integer solutions are there to $y^2=x^2-2017$?

[b]p1.[/b] What is the smallest positive integer divisible by $20$, $12$, and $13$? [b]p2.[/b] Two circles of radius $5$ are placed in the plane such that their centers are $7$ units apart. What is the largest possible distance between a point on one circle and a point on the other? [b]p3.[/b] In a magic square, all the numbers in the rows, columns, and diagonals sum to the same value. How many $2\times 2$ magic squares containing the integers $\{1, 2, 3, 4\}$ are there? [b]p4.[/b] Ethan's sock drawer contains two pairs of white socks and one pair of red socks. Ethan picks two socks at random. What is the probability that he picks two white socks? [b]p5.[/b] The sum of the time on a digital clock is the sum of the digits displayed on the screen. For example, the sum of the time at $10:23$ would be $6$. Assuming the clock is a $12$ hour clock, what is the greatest possible positive difference between the sum of the time at some time and the sum of the time one minute later? [b]p6.[/b] Given the expression $1 \div 2 \div 3 \div 4$, what is the largest possible resulting value if one were to place parentheses $()$ somewhere in the expression? [b]p7.[/b] At a convention, there are many astronomers, astrophysicists, and cosmologists. At $first$, all the astronomers and astrophysicists arrive. At this point, $\frac35$ of the people in the room are astronomers. Then, all the cosmologists come, so now, $30\%$ of the people in the room are astrophysicists. What fraction of the scientists are cosmologists? [b]p8.[/b] At $10:00$ AM, a minuteman starts walking down a $1200$-step stationary escalator at $40$ steps per minute. Halfway down, the escalator starts moving up at a constant speed, while the minuteman continues to walk in the same direction and at the same pace that he was going before. At $10:55$ AM, the minuteman arrives back at the top. At what speed is the escalator going up, in steps per minute? [b]p9.[/b] Given that $x_1 = 57$, $x_2 = 68$, and $x_3 = 32$, let $x_n = x_{n-1} -x_{n-2} +x_{n-3}$ for $n \ge 4$. Find $x_{2013}$. [b]p10.[/b] Two squares are put side by side such that one vertex of the larger one coincides with a vertex of the smaller one. The smallest rectangle that contains both squares is drawn. If the area of the rectangle is $60$ and the area of the smaller square is $24$, what is the length of the diagonal of the rectangle? [b]p11.[/b] On a dield trip, $2$ professors, $4$ girls, and $4$ boys are walking to the forest to gather data on butterflies. They must walk in a line with following restrictions: one adult must be the first person in the line and one adult must be the last person in the line, the boys must be in alphabetical order from front to back, and the girls must also be in alphabetical order from front to back. How many such possible lines are there, if each person has a distinct name? [b]p12.[/b] Flatland is the rectangle with vertices $A, B, C$, and $D$, which are located at $(0, 0)$, $(0, 5)$, $(5, 5)$, and $(5, 0)$, respectively. The citizens put an exact map of Flatland on the rectangular region with vertices $(1, 2)$, $(1, 3)$, $(2, 3)$, and $(2, 2)$ in such a way so that the location of $A$ on the map lies on the point $(1, 2)$ of Flatland, the location of $B$ on the map lies on the point $(1, 3)$ of Flatland, the location of C on the map lies on the point $(2, 3)$ of Flatland, and the location of D on the map lies on the point $(2, 2)$ of Flatland. Which point on the coordinate plane is thesame point on the map as where it actually is on Flatland? [b]p13.[/b] $S$ is a collection of integers such that any integer $x$ that is present in $S$ is present exactly $x$ times. Given that all the integers from $1$ through $22$ inclusive are present in $S$ and no others are, what is the average value of the elements in $S$? [b]p14.[/b] In rectangle $PQRS$ with $PQ < QR$, the angle bisector of $\angle SPQ$ intersects $\overline{SQ}$ at point $T$ and $\overline{QR }$ at $U$. If $PT : TU = 3 : 1$, what is the ratio of the area of triangle $PTS$ to the area of rectangle $PQRS$? [b]p15.[/b] For a function $f(x) = Ax^2 + Bx + C$, $f(A) = f(B)$ and $A + 6 = B$. Find all possible values of $B$. [b]p16.[/b] Let $\alpha$ be the sum of the integers relatively prime to $98$ and less than $98$ and $\beta$ be the sum of the integers not relatively prime to $98$ and less than $98$. What is the value of $\frac{\alpha}{\beta}$ ? [b]p17.[/b] What is the value of the series $\frac{1}{3} + \frac{3}{9} + \frac{6}{27} + \frac{10}{81} + \frac{15}{243} + ...$? [b]p18.[/b] A bug starts at $(0, 0)$ and moves along lattice points restricted to $(i, j)$, where $0 \le i, j \le 2$. Given that the bug moves $1$ unit each second, how many different paths can the bug take such that it ends at $(2, 2)$ after $8$ seconds? [b]p19.[/b] Let $f(n)$ be the sum of the digits of $n$. How many different values of $n < 2013$ are there such that $f(f(f(n))) \ne f(f(n))$ and $f(f(f(n))) < 10$? [b]p20.[/b] Let $A$ and $B$ be points such that $\overline{AB} = 14$ and let $\omega_1$ and $\omega_2$ be circles centered at $A$ and $B$ with radii $13$ and $15$, respectively. Let $C$ be a point on $\omega_1$ and $D$ be a point on $\omega_2$ such that $\overline{CD}$ is a common external tangent to $\omega_1$ and $\omega_2$. Let $P$ be the intersection point of the two circles that is closer to $\overline{CD}$. If $M$ is the midpoint of $\overline{CD}$, what is the length of segment $\overline{PM}$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^+, A^-, B^+, B^-, C^+$, and $C^-$ is rolled. Suppose the bee occupies the point $(a, b, c)$. If the die shows $A^+$, then the bee moves to the point $(a+1, b, c)$ and if the die shows $A^-$, then the bee moves to the point $(a-1, b, c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube? $ \textbf{(A) }\frac{1}{54} \qquad \textbf{(B) }\frac{7}{54} \qquad \textbf{(C) }\frac{1}{6} \qquad \textbf{(D) }\frac{5}{18} \qquad \textbf{(E) }\frac{2}{5} \qquad $

A meeting was attended by $n$ people. They are welcome to occupy the $k$ table provided $\left( k \le \frac{n}{2} \right)$. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that $A$ is the number of ways to choose representatives to speak. Determine the maximum value of $A$ that is possible.

Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.

Show that if the series $$\sum_{n=1}^{\infty} \frac{1}{p_n}$$ is convergent, where $p_1,p_2,p_3,\dots, p_n, \dots$ are positive real numbers, then the series $$\sum_{n=1}^{\infty} \frac{n^2}{(p_1+p_2+\dots +p_n)^2}p_n$$ is also convergent.

Let $ABC$ be an isosceles triangle with $AB = AC$. A semi-circle of diameter $[EF] $ with $E, F \in [BC]$, is tangent to the sides $AB,AC$ in $M, N$ respectively and $AE$ intersects the semicircle at $P$. Prove that $PF$ passes through the midpoint of $[MN]$.

Let $n\ge 3$ be an integer and let $A_{1}, A_{2},\dots, A_{n}$ be $n$ distinct subsets of $S=\{1, 2,\dots, n\}$. Show that there exists $x\in S$ such that the n subsets $A_{i}-\{x\}, i=1,2,\dots n$ are also disjoint. what i have is [hide="this"]we may assume that the union of the $A_{i}$s is $S$. [/hide]

Compute the number of ordered pairs $(a,b)$ of positive integers satisfying $a^b=2^{100}$. [i]Proposed by Nathan Xiong[/i]

Find all four-digit natural numbers formed by two even digits and two odd digits that verify that when multiplied by $2$ four-digit numbers are obtained with all their even digits and when divided by $2$ four-digit natural numbers are obtained with all their odd digits.

An acute non-isosceles $\triangle ABC$ is given. $CD, AE, BF$ are its altitudes. The points $E', F'$ are symetrical of $E, F$ with respect accordingly to $A$ and $B$. The point $C_1$ lies on $\overrightarrow{CD}$, such that $DC_1=3CD$. Prove that $\angle E'C_1F'=\angle ACB$

For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.

There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does? [i]Mikhail Svyatlovskiy[/i]

The triangle $ \triangle SFA$ has a right angle at $ F$. The points $ P$ and $ Q$ lie on the line $ SF$ such that the point $ P$ lies between $ S$ and $ F$ and the point $ F$ is the midpoint of the segment $ [PQ]$. The circle $ k_1$ is th incircle of the triangle $ \triangle SPA$. The circle $ k_2$ lies outside the triangle $ \triangle SQA$ and touches the segment $ [QA]$ and the lines $ SQ$ and $ SA$. Prove that the sum of the radii of the circles $ k_1$ and $ k_2$ equals the length of $ [FA]$.

[b]p21[/b]. If $f = \cos(\sin (x))$. Calculate the sum $\sum^{2021}_{n=0} f'' (n \pi)$. [b]p22.[/b] Find all real values of $A$ that minimize the difference between the local maximum and local minimum of $f(x) = \left(3x^2 - 4\right)\left(x - A + \frac{1}{A}\right)$. [b]p23.[/b] Bessie is playing a game. She labels a square with vertices labeled $A, B, C, D$ in clockwise order. There are $7$ possible moves: she can rotate her square $90$ degrees about the center, $180$ degrees about the center, $270$ degrees about the center; or she can flip across diagonal $AC$, flip across diagonal $BD$, flip the square horizontally (flip the square so that vertices A and B are switched and vertices $C$ and $D$ are switched), or flip the square vertically (vertices $B$ and $C$ are switched, vertices $A$ and $D$ are switched). In how many ways can Bessie arrive back at the original square for the first time in $3$ moves? [b]p24.[/b] A positive integer is called [i]happy [/i] if the sum of its digits equals the two-digit integer formed by its two leftmost digits. Find the number of $5$-digit happy integers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].