Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations \[x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n\] where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.

Let $\mathbb{N}$ be the set of all positive integers. $f,g:\mathbb{N} \to \mathbb{N}$ be funtions such that $f$ is onto and $g$ is one-one and $f(n)\geq g(n)$ for all positive integers $n$. Prove that $f=g$.

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$. [i]Merlijn Staps[/i]

Let $ABCD$ be a convex quadrilateral with $\angle BCD= 120^o, \angle {CBA} = 45^o, \angle {CBD} = 15^o$ and $\angle {CAB} = 90^o$. Show that $AB = AD$.

[b][u]BdMO National Higher Secondary Problem 3[/u][/b] Let $N$ be the number if pairs of integers $(m,n)$ that satisfies the equation $m^2+n^2=m^3$ Is $N$ finite or infinite?If $N$ is finite,what is its value?

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

Let $A=\{ x\in \mathbb{N},x=a^2+2b^2,a,b\in \mathbb{Z},ab\neq 0 \}$ and $p$ is a prime number. Prove that if $p^2\in A$, then $p\in A$.

Do there exist two polygons such that, by putting them together in three different ways (without holes, overlap, or reflections), we can obtain first a triangle, then a convex quadrilateral, and lastly a convex pentagon?

Let incircle $(I)$ of triangle $ABC$ touch the sides $BC,CA,AB$ at $D,E,F$ respectively. Let $(O)$ be the circumcircle of $ABC$. Ray $EF$ meets $(O)$ at $M$. Tangents at $M$ and $A$ of $(O)$ meet at $S$. Tangents at $B$ and $C$ of $(O)$ meet at $T$. Line $TI$ meets $OA$ at $J$. Prove that $\angle ASJ=\angle IST$.

Among $2000$ outwardly indistinguishable balls, wines - aluminum weighing 1$0$ g, and the rest - duralumin weighing $9.9$ g. It is required to select two piles of balls so that the masses of the piles are different, and the number of balls in them - the same. What is the smallest number of weighings on a cup scale without weights that can be done?

Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices?

Alison is eating $2401$ grains of rice for lunch. She eats the rice in a very peculiar manner: every step, if she has only one grain of rice remaining, she eats it. Otherwise, she nds the smallest positive integer $d > 1$ for which she can group the rice into equal groups of size $d$ with none left over. She then groups the rice into groups of size $d$, eats one grain from each group, and puts the rice back into a single pile. How many steps does it take her to finish all her rice?

Can we find a subset $A$ of $\mathbb{N}$ containing exactly five numbers such that sum of any three elements of $A$ is a prime number? Justify your answer.

Let $n$ be a positive integer.Consider the polynomial $p(x)=x^2+x+1$. What is the remainder of $ x^3$ when divided by $x^2+x+1$.For what positive integers values of $n$ is $ x^{2n}+x^n+1$ divisible by $p(x)$? Post no:[size=300]$100$[/size]

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

In a triangle $ABC$, $I$ and $O$ are the incenter and circumcenter respectively, $A',B',C'$ the excenters, and $O'$ the circumcenter of $\triangle A'B'C'$. If $R$ and $R'$ are the circumradii of triangles $ABC$ and $A'B'C'$, respectively, prove that: (i) $R'= 2R $ (ii) $IO' = 2IO$

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?

Prove that for every natural number \( n>2 \) there exists an integer \( k \) that can be written as the sum of \( i \) positive perfect squares, for every \( i \) between \( 2 \) and \( n \).

In a trapezoid $ABCD$, the side $DA$ is perpendicular to the bases $AB$ and $CD$. Also $AB=45$, $CD =20$, $BC =65$. Let $P$ be a point on the side $BC$ such that $BP=45$ and let $M$ be the midpoint of $DA$. Calculate the length of $PM$ .

There are three types of tiles: an L-shaped tile with three $1\times 1$ squares, a $2\times 2$ square, and a Z-shaped tile with four $1\times 1$ squares. We tile a $(2n-1)\times (2n-1)$ square using these tiles. Prove that there are at least $4n-1$ L-shaped tiles. I'm sorry about my poor description, but I don't know how to draw pictures...

$N^3$ unit cubes are made into beads by drilling a hole through them along a diagonal, put on a string and binded. Thus the cubes can move freely in space as long as the vertices of two neighboring cubes (including the first and last one) are touching. For which $N$ is it possible to build a cube of edge $N$ using these cubes?

[b]p1.[/b] What is the largest number of five dollar footlongs Jimmy can buy with 88 dollars? [b]p2.[/b] Austin, Derwin, and Sylvia are deciding on roles for BMT $2021$. There must be a single Tournament Director and a single Head Problem Writer, but one person cannot take on both roles. In how many ways can the roles be assigned to Austin, Derwin, and Sylvia? [b]p3.[/b] Sofia has$ 7$ unique shirts. How many ways can she place $2$ shirts into a suitcase, where the order in which Sofia places the shirts into the suitcase does not matter? [b]p4.[/b] Compute the sum of the prime factors of $2021$. [b]p5.[/b] A sphere has volume $36\pi$ cubic feet. If its radius increases by $100\%$, then its volume increases by $a\pi$ cubic feet. Compute $a$. [b]p6.[/b] The full price of a movie ticket is $\$10$, but a matinee ticket to the same movie costs only $70\%$ of the full price. If $30\%$ of the tickets sold for the movie are matinee tickets, and the total revenue from movie tickets is $\$1001$, compute the total number of tickets sold. [b]p7.[/b] Anisa rolls a fair six-sided die twice. The probability that the value Anisa rolls the second time is greater than or equal to the value Anisa rolls the first time can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p8.[/b] Square $ABCD$ has side length $AB = 6$. Let point $E$ be the midpoint of $\overline{BC}$. Line segments $\overline{AC}$ and $\overline{DE}$ intersect at point $F$. Compute the area of quadrilateral ABEF. [b]p9.[/b] Justine has a large bag of candy. She splits the candy equally between herself and her $4$ friends, but she needs to discard three candies before dividing so that everyone gets an equal number of candies. Justine then splits her share of the candy between herself and her two siblings, but she needs to discard one candy before dividing so that she and her siblings get an equal number of candies. If Justine had instead split all of the candy that was originally in the large bag between herself and $14$ of her classmates, what is the fewest number of candies that she would need to discard before dividing so that Justine and her $14$ classmates get an equal number of candies? [b]p10.[/b] For some positive integers $a$ and $b$, $a^2 - b^2 = 400$. If $a$ is even, compute $a$. [b]p11.[/b] Let $ABCDEFGHIJKL$ be the equilateral dodecagon shown below, and each angle is either $90^o$ or $270^o$. Let $M$ be the midpoint of $\overline{CD}$, and suppose $\overline{HM}$ splits the dodecagon into two regions. The ratio of the area of the larger region to the area of the smaller region can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/387bcdf2a6c39fcada4f21f24ceebd18a7f887.png[/img] [b]p12.[/b] Nelson, who never studies for tests, takes several tests in his math class. Each test has a passing score of $60/100$. Since Nelson's test average is at least $60/100$, he manages to pass the class. If only nonnegative integer scores are attainable on each test, and Nelson gets a dierent score on every test, compute the largest possible ratio of tests failed to tests passed. Assume that for each test, Nelson either passes it or fails it, and the maximum possible score for each test is 100. [b]p13.[/b] For each positive integer $n$, let $f(n) = \frac{n}{n+1} + \frac{n+1}{n}$ . Then $f(1)+f(2)+f(3)+...+f(10)$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p14.[/b] Triangle $\vartriangle ABC$ has point $D$ lying on line segment $\overline{BC}$ between $B$ and $C$ such that triangle $\vartriangle ABD$ is equilateral. If the area of triangle $\vartriangle ADC$ is $\frac14$ the area of triangle $\vartriangle ABC$, then $\left( \frac{AC}{AB}\right)^2$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [b]p15.[/b] In hexagon $ABCDEF$, $AB = 60$, $AF = 40$, $EF = 20$, $DE = 20$, and each pair of adjacent edges are perpendicular to each other, as shown in the below diagram. The probability that a random point inside hexagon $ABCDEF$ is at least $20\sqrt2$ units away from point $D$ can be expressed in the form $\frac{a-b\pi}{c}$ , where $a$, $b$, $c$ are positive integers such that gcd$(a, b, c) = 1$. Compute $a + b + c$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/1b45470265d10a73de7b83eff1d3e3087d6456.png[/img] [b]p16.[/b] The equation $\sqrt{x} +\sqrt{20-x} =\sqrt{20 + 20x - x^2}$ has $4$ distinct real solutions, $x_1$, $x_2$, $x_3$, and $x_4$. Compute $x_1 + x_2 + x_3 + x_4$. [b]p17.[/b] How many distinct words with letters chosen from $B, M, T$ have exactly $12$ distinct permutations, given that the words can be of any length, and not all the letters need to be used? For example, the word $BMMT$ has $12$ permutations. Two words are still distinct even if one is a permutation of the other. For example, $BMMT$ is distinct from $TMMB$. [b]p18.[/b] We call a positive integer binary-okay if at least half of the digits in its binary (base $2$) representation are $1$'s, but no two $1$s are consecutive. For example, $10_{10} = 1010_2$ and $5_{10} = 101_2$ are both binary-okay, but $16_{10} = 10000_2$ and $11_{10} = 1011_2$ are not. Compute the number of binary-okay positive integers less than or equal to $2020$ (in base $10$). [b]p19.[/b] A regular octahedron (a polyhedron with $8$ equilateral triangles) has side length $2$. An ant starts on the center of one face, and walks on the surface of the octahedron to the center of the opposite face in as short a path as possible. The square of the distance the ant travels can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/f/8/3aa6abe02e813095e6991f63fbcf22f2e0431a.png[/img] [b]p20.[/b] The sum of $\frac{1}{a}$ over all positive factors $a$ of the number $360$ can be expressed in the form $\frac{m}{n}$ ,where $m$ and $n$ are relatively prime positive integers. Compute $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]