The angles of a pentagon are in arithmetic progression. One of the angles in degrees, must be: $ \textbf{(A)}\ 108 \qquad \textbf{(B)}\ 90 \qquad \textbf{(C)}\ 72 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 36$

Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.

For a positive integer $n$ and real numbers $a_1, a_2, \dots ,a_n$ we'll define $b_1, b_2, \dots ,b_{n+1}$ such that $b_k=a_k+\max({a_{k+1},a_{k+2}})$ for all $1\leq k \leq n$ and $b_{n+1}=b_1$. (Also $a_{n+1}=a_1$ and $a_{n+2}=a_2$) Find the least possible value of $\lambda$ such that for all $n, a_1, \dots, a_n$ the inequality $$\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}$$ holds.

Let $ [a]$ be the integer such that $ [a]\le a<[a]\plus{}1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}\plus{}[b]\plus{}\{c\}\equal{}2.9\\\{b\}\plus{}[c]\plus{}\{a\}\equal{}5.3\\\{c\}\plus{}[a]\plus{}\{b\}\equal{}4.0.\]

Does there exist a function $f : \mathbb N \to \mathbb N$, such that $f(f(n)) =n + 1987$ for every natural number $n$? [i](IMO Problem 4)[/i] [i]Proposed by Vietnam.[/i]

Construct a convex polygon such that each of its sides has the same length as one of its diagonals and each diagonal has the same length as one of its sides, or prove that such a polygon does not exist.

Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies \[ f^n(n) = n + k \] for all $n \in \mathbb{N}$

Yatta and Yogi play a game in which they begin with a pile of $n$ stones. The players take turns removing $1$, $2$, $3$, $5$, $6$, $7$, or $8$ stones from the pile. That is, when it is a player's turn to remove stones, that player may remove from $1$ to $8$ stones, but [i]cannot[/i] remove exactly $4$ stones. The player who removes the last stone [i]loses[/i]. Yogi goes first and finds that he has a winning position, meaning that so long as he plays perfectly, Yatta cannot defeat him. For how many positive integers $n$ from $100$ to $2008$ inclusive is this the case?

Let $ABC$ be an isosceles triangle $(AC = BC)$, $O$ be its circumcenter, $H$ be the orthocenter, and $P$ be a point inside the triangle such that $\angle APH = \angle BPO = \pi /2$. Prove that $\angle PAC = \angle PBA = \angle PCB$.

Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.

Find all functions $f:R\to R$ such that for all real $x,y$ with $y\ne 0$ $$f(x-f(x/y))=xf(1-f(1/y))$$ and a) $f(1-f(1))\ne 0$ b) $ f(1-f(1))= 0$ S. Kuzmich, I.Voronovich

Prove that for any natural number $n < 2310 $ , $n(2310-n)$ is not divisible by $2310$.

It is given rectangle $ABCD$ with length $|AB|=15cm$ and with length of altitude $|BE|=12cm$ where $BC$ is altitude of triangle $ABC$ . Find perimeter and area of rectangle $ABCD$ .

Let $O$ be the intersection point of medians $AP$ and $CQ$ of triangle $ABC$. If $OQ$ is $3$ inches, then $OP$, in inches, is: $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \dfrac{9}{2} \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \text{undetermined}$

Kiana has two older twin brothers. The product of their ages is $ 128$. What is the sum of their three ages? $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 24$

A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.

Let $ABC$ be a triangle with circumcenter $O$. Point $D, E, F$ are chosen on sides $AB, BC$ and $AC$, respectively, such that $ADEF$ is a rhombus. The circumcircles of $BDE$ and $CFE$ intersect $AE$ at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Ariel García[/i]

Find the minimu value of $\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \{x\cos t+(1-x)\sin t\}^2dt.$ [i]2010 Ibaraki University entrance exam/Science[/i]

[b]p1.[/b] The clock is $2$ hours $20$ minutes ahead of the correct time each week. The clock is set to the correct time at midnight Sunday to Monday. What time does this clock show at 6pm correct time on Thursday? [b]p2.[/b] Five cities $A,B,C,D$, and $E$ are located along the straight road in the alphabetical order. The sum of distances from $B$ to $A,C,D$ and $E$ is $20$ miles. The sum of distances from $C$ to the other four cities is $18$ miles. Find the distance between $B$ and $C$. [b]p3.[/b] Does there exist distinct digits $a, b, c$, and $d$ such that $\overline{abc}+\overline{c} = \overline{bda}$? Here $\overline{abc}$ means the three digit number with digits $a, b$, and $c$. [b]p4.[/b] Kuzya, Fyokla, Dunya, and Senya participated in a mathematical competition. Kuzya solved $8$ problems, more than anybody else. Senya solved $5$ problem, less than anybody else. Each problem was solved by exactly $3$ participants. How many problems were there? [b]p5.[/b] Mr Mouse got to the cellar where he noticed three heads of cheese weighing $50$ grams, $80$ grams, and $120$ grams. Mr. Mouse is allowed to cut simultaneously $10$ grams from any two of the heads and eat them. He can repeat this procedure as many times as he wants. Can he make the weights of all three pieces equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a $n$-sided regular polygon, $n \geq 4$, and let $V$ be a subset of $r$ vertices of the polygon. Show that if $r(r-3) \geq n$, then there exist at least two congruent triangles whose vertices belong to $V$.

[b]p1.[/b] What is $2 \cdot 24 + 20 \cdot 24 + 202 \cdot 4 + 2024$? [b]p2.[/b] Jerry has $300$ legos. Tie can either make cars, which require $17$ legos, or bikes, which require $13$ legos. Assuming he uses all of his legos, how many ordered pairs $(a, b)$ are there such that he makes $a$ cars and $b$ bikes? [b]p3.[/b] Patrick has $7$ unique textbooks: $2$ Geometry books, $3$ Precalculus books and $2$ Algebra II books. How many ways can he arrange his books on a bookshelf such that all the books of the same subjects are adjacent to each other? [b]p4.[/b] After a hurricane, a $32$ meter tall flagpole at the Act on-Boxborough Regional High School snapped and fell over. Given that the snapped part remains in contact with the original pole, and the top of the polo falls $24$ meters away from the bottom of the pole, at which height did the polo snap? (Assume the flagpole is perpendicular to the ground.) [b]p5.[/b] Jimmy is selling lemonade. Iio has $200$ cups of lemonade, and he will sell them all by the end of the day. Being the ethically dubious individual he is, Jimmy intends to dilute a few of the cups of lemonade with water to conserve resources. Jimmy sells each cup for $\$4$. It costs him $\$ 1$ to make a diluted cup of lemonade, and it costs him $\$2.75$ to make a cup of normal lemonade. What is the minimum number of diluted cups Jimmy must sell to make a profit of over $\$400$? [b]p6.[/b] Jeffrey has a bag filled with five fair dice: one with $4$ sides, one with $6$ sides, one with $8$ sides, one with $12$ sides, and one with $20$ sides. The dice are numbered from $1$ to the number of sides on the die. Now, Marco will randomly pick a die from .Jeffrey's bag and roll it. The probability that Marco rolls a $7$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$. [b]p7.[/b] What is the remainder when the sum of the first $2024$ odd numbers is divided by $6072$? [b]p8.[/b] A rhombus $ABCD$ with $\angle A = 60^o$ and $AB = 600$ cm is drawn on a piece of paper. Three ants start moving from point $A$ to the three other points on the rhombus. One ant walks from $A$ to $B$ at a leisurely speed of $10$ cm/s. The second ant runs from $A$ to $C$ at a slightly quicker pace of $6\sqrt3$ cm/s, arriving to $C$ $x$ seconds after the first ant. The third ant travels from $A$ to $B$ to $D$ at a constant speed, arriving at $D$ $x$ seconds after the second ant. The speed of the last ant can be written as $\frac{m}{n}$ cm/s, where $m$ and $n$ are relatively prime positive integers. Find $mn$. [b]p9.[/b] This year, the Apple family has harvested so many apples that they cannot sell them all! Applejack decides to make $40$ glasses of apple cider to give to her friends. If Twilight and Fluttershy each want $1$ or $2$ glasses; Pinkie Pic wants cither $2$, $14$, or $15$ glasses; Rarity wants an amount of glasses that is a power of three; and Rainbow Dash wants any odd number of glasses, then how many ways can Applejack give her apple cider to her friends? Note: $1$ is considered to be a power of $3$. [b]p10.[/b] Let $g_x$ be a geometric sequence with first term $27$ and successive ratio $2n$ (so $g_{x+1}/g_x = 2n$). Then, define a function $f$ as $f(x) = \log_n(g_x)$, where $n$ is the base of the logarithm. It is known that the sum of the first seven terms of $f(x)$ is $42$. Find $g_2$, the second term of the geometric sequence. Note: The logarithm base $b$ of $x$, denoted $\log_b(x)$ is equal to the value $y$ such that $b^y = x$. In other words, if $\log_b(x) = y$, then $b^y = x$. [b]p11.[/b] Let $\varepsilon$ be an ellipse centered around the origin, such that its minor axis is perpendicular to the $x$-axis. The length of the ellipse's major and minor axes is $8$ and $6$, respectively. Then, let $ABCD$ be a rectangle centered around the origin, such that $AB$ is parallel to the $x$-axis. The lengths of $AB$ and $BC$ are $8$ and $3\sqrt2$, respectively. The area outside the ellipse but inside the rectangle can be expressed as $a\sqrt{b}-c-d\pi$, for positive integers $a$, $b$, $c$, $d$ where $b$ is not divisible by a perfect square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/e/c/9d943966763ee7830d037ef98c21139cf6f529.png[/img] [b]p12.[/b] Let $N = 2^7 \cdot 3^7 \cdot 5^5$. Find the number of ways to express $N$ as the product of squares and cubes, all of which are integers greater than $1$. [b]p13.[/b] Jerry and Eric are playing a $10$-card game where Jerry is deemed the ’’landlord" and Eric is deemed the ' peasant'’. To deal the cards, the landlord keeps one card to himself. Then, the rest of the $9$ cards are dealt out, such that each card has a $1/2$ chance to go to each player. Once all $10$ cards are dealt out, the landlord compares the number of cards he owns with his peasant. The probability that the landlord wins is the fraction of cards he has. (For example, if Jerry has $5$ cards and Eric has $2$ cards, Jerry has a$ 5/7$ ths chance of winning.) The probability that Jerry wins the game can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Find $p + q$. [b]p14.[/b] Define $P(x) = 20x^4 + 24x^3 + 10x^2 + 21x+ 7$ to have roots $a$, $b$, $c$, and $d$. If $Q(x)$ has roots $\frac{1}{a-2}$,$\frac{1}{b-2}$,$ \frac{1}{c-2}$, $\frac{1}{d-2}$ and integer coefficients with a greatest common divisor of $1$, then find $Q(2)$. [b]p15.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 14$, $BC = 13$, and $AC = 15$. The incircle of $\vartriangle ABC$ is drawn with center $I$, tangent to $\overline{AB}$ at $X$. The line $\overleftrightarrow{IX}$ intersects the incircle again at $Y$ and intersects $\overline{AC}$ at $Z$. The area of $\vartriangle AYZ$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the $1000$ th digit to the right of the decimal point in the decimal representation of $\tfrac{37}{5500}$?

For an integer $m\ge 3$, let $S(m)=1+\frac{1}{3}+…+\frac{1}{m}$ (the fraction $\frac12$ does not participate in addition and does participate in fractions $\frac{1}{k}$ for integers from $3$ until $m$). Let $n\ge 3$ and $ k\ge 3$ . Compare the numbers $S(nk)$ and $S(n)+S(k)$ .

Let $A_1 , A_2 ,\ldots, A_{1066}$ be subsets of a finite set $X$ such that $|A_i | > \frac{1}{2} |X|$ for $1\leq i \leq 1066.$ Prove that there exist ten elements $x_1 ,x_2 ,\ldots , x_{10}$ of $X$ such that every $A_i $ contains at least one of $x_1 , x_2 ,\ldots, x_{10}.$

Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\] Holds. [i] Nikolai Nikolov, Oleg Mushkarov[/i]