One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

Let $U$ and $C$ be two circles, and kite $BERK$ have vertices that lie on $U$ and sides that are tangent to $C.$ Given that the diagonals of the kite measure $5$ and $6,$ find the ratio of the area of $U$ to the area of $C.$

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , find $a + b$.

Which of the following numbers is [b]not[/b] a perfect square? $\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}$

For a positive integer $n$ and a subset $S$ of $\{1, 2, \dots, n\}$, let $S$ be "$n$-good" if and only if for any $x$, $y\in S$ (allowed to be same), if $x+y\leq n$, then $x+y\in S$. Let $r_n$ be the smallest real number such that for any positive integer $m\leq n$, there is always a $m$-element "$n$-good" set, so that the sum of its elements is not more than $m\cdot r_n$. Prove that there exists a real number $\alpha$ such that for any positive integer $n$, $|r_n-\alpha n|\leq 2024.$

Inscribed circle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. Let $AA_{1}$, $BB_{1}$ and $CC_{1}$ be the altitudes of the triangle $ABC$ and $M$, $N$ and $P$ be the incenters of triangles $AB_{1}C_{1}$, $BC_{1}A_{1}$ and $CA_{1}B_{1}$, respectively. a) Prove that $M$, $N$ and $P$ are orthocentres of triangles $AYZ$, $BZX$ and $CXY$, respectively. b) Prove that common external tangents of these incircles, different from triangle sides, are concurent at orthocentre of triangle $XYZ$.

For how many nonnegative integer values of $k$ does the equation $7x^2 +kx +11 = 0$ have no real solutions?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

We wish to place natural numbers around a circle such that the following property is satisfied: the absolute values of the differences of each pair of neighboring numbers are all different. a) Is it possible to place the numbers from 1 to 2009 satisfying this property b) Is it possible to suppress one of the numbers from 1 to 2009 in such a way that the remaining 2008 numbers can be placed satisfying the property

[u]Set 1[/u] [b]p1.[/b] Find $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11$. [b]p2.[/b] Find $1 \cdot 11 + 2 \cdot 10 + 3 \cdot 9 + 4 \cdot 8 + 5 \cdot 7 + 6 \cdot 6$. [b]p3.[/b] Let $\frac{1}{1\cdot 2} +\frac{1}{2\cdot 3} +\frac{1}{3\cdot 4} +\frac{1}{4\cdot 5} +\frac{1}{5\cdot 6} +\frac{1}{6\cdot 7} +\frac{1}{7\cdot 8} +\frac{1}{8\cdot 9} +\frac{1}{9\cdot 10} +\frac{1}{10\cdot 11} =\frac{m}{n}$ , where $m$ and $n$ are positive integers that share no prime divisors. Find $m + n$. [u]Set 2[/u] [b]p4.[/b] Define $0! = 1$ and let $n! = n \cdot (n - 1)!$ for all positive integers $n$. Find the value of $(2! + 0!)(1! + 8!)$. [b]p5.[/b] Rachel’s favorite number is a positive integer $n$. She gives Justin three clues about it: $\bullet$ $n$ is prime. $\bullet$ $n^2 - 5n + 6 \ne 0$. $\bullet$ $n$ is a divisor of $252$. What is Rachel’s favorite number? [b]p6.[/b] Shen eats eleven blueberries on Monday. Each day after that, he eats five more blueberries than the day before. For example, Shen eats sixteen blueberries on Tuesday. How many blueberries has Shen eaten in total before he eats on the subsequent Monday? [u]Set 3[/u] [b]p7.[/b] Triangle $ABC$ satisfies $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ can be expressed in the form $m\sqrt{n}$, where $n$ is not divisible by the square of a prime, then determine $m + n$. [b]p8.[/b] Sebastian is playing the game Split! on a coordinate plane. He begins the game with one token at $(0, 0)$. For each move, he is allowed to select a token on any point $(x, y)$ and take it off the plane, replacing it with two tokens, one at $(x + 1, y)$, and one at $(x, y + 1)$. At the end of the game, for a token on $(a, b)$, it is assigned a score $\frac{1}{2^{a+b}}$ . These scores are summed for his total score. Determine the highest total score Sebastian can get in $100$ moves. [b]p9.[/b] Find the number of positive integers $n$ satisfying the following two properties: $\bullet$ $n$ has either four or five digits, where leading zeros are not permitted, $\bullet$ The sum of the digits of $n$ is a multiple of $3$. [u]Set 4[/u] [b]p10.[/b] [i]A unit square rotated $45^o$ about a vertex, Sweeps the area for Farmer Khiem’s pen. If $n$ is the space the pigs can roam, Determine the floor of $100n$.[/i] If $n$ is the area a unit square sweeps out when rotated 4$5$ degrees about a vertex, determine $\lfloor 100n \rfloor$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/129efd0dbd56dc0b4fb742ac80eaf2447e106d.png[/img] [b]p11.[/b][i] Michael is planting four trees, In a grid, three rows of three, If two trees are close, Then both are bulldozed, So how many ways can it be?[/i] In a three by three grid of squares, determine the number of ways to select four squares such that no two share a side. [b]p12.[/b] [i]Three sixty-seven Are the last three digits of $n$ cubed. What is $n$?[/i] If the last three digits of $n^3$ are $367$ for a positive integer $n$ less than $1000$, determine $n$. [u]Set 5[/u] [b]p13.[/b] Determine $\sqrt[4]{97 + 56\sqrt{3}} + \sqrt[4]{97 - 56\sqrt{3}}$. [b]p14. [/b]Triangle $\vartriangle ABC$ is inscribed in a circle $\omega$ of radius $12$ so that $\angle B = 68^o$ and $\angle C = 64^o$ . The perpendicular from $A$ to $BC$ intersects $\omega$ at $D$, and the angle bisector of $\angle B$ intersects $\omega$ at $E$. What is the value of $DE^2$? [b]p15.[/b] Determine the sum of all positive integers $n$ such that $4n^4 + 1$ is prime. [u]Set 6[/u] [b]p16.[/b] Suppose that $p, q, r$ are primes such that $pqr = 11(p + q + r)$ such that $p\ge q \ge r$. Determine the sum of all possible values of $p$. [b]p17.[/b] Let the operation $\oplus$ satisfy $a \oplus b =\frac{1}{1/a+1/b}$ . Suppose $$N = (...((2 \oplus 2) \oplus 2) \oplus ... 2),$$ where there are $2018$ instances of $\oplus$ . If $N$ can be expressed in the form $m/n$, where $m$ and $n$ are relatively prime positive integers, then determine $m + n$. [b]p18.[/b] What is the remainder when $\frac{2018^{1001} - 1}{2017}$ is divided by $2017$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777307p24369763]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a, b, c, \lambda$ be positive real numbers with $\lambda \geq 1/4$. Show that $$\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.$$

Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

Let $ABCD$ be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at $H$. Denote by $M$ and $N$ the midpoints of $\overline{BC}$ and $\overline{CD}$. Rays $MH$ and $NH$ meet $\overline{AD}$ and $\overline{AB}$ at $S$ and $T$, respectively. Prove that there exists a point $E$, lying outside quadrilateral $ABCD$, such that [list] [*] ray $EH$ bisects both angles $\angle BES$, $\angle TED$, and [*] $\angle BEN = \angle MED$. [/list] [i]Proposed by Evan Chen[/i]

The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star. [img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]

Find all natural numbers $x,y,z$, such that $7^{x}+13^{y}=2^{z}$.

Let \[f(x)=x^4-6x^3+26x^2-46x+65.\] Let the roots of $f(x)$ be $a_k+ib_k$ for $k=1,2,3,4$. Given that the $a_k$, $b_k$ are all integers, find $|b_1|+|b_2|+|b_3|+|b_4|.$

Let $M$ be a compact orientable $2n$-manifold with boundary, where $n \ge 2$. Suppose that $H_0(M;\mathbb{Q}) \cong \mathbb{Q}$ and $H_i(M;\mathbb{Q})=0$ for $i>0$. Prove that the order of $H_{n-1}(\partial M; \mathbb{Z})$ is a square number.

Prove that for all positive real numbers $x, y$ holds the inequality $$x^4 + y^3 + x^2 + y + 1 > \frac92 xy.$$

Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that: $ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and $ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$.

Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$

Let $ABC$ be a right triangle at $B$. Let $D$ be a point such that $BD\perp AC$ and $DC = AC$. Find the ratio $\frac{AD}{AB}$.

A circle $ C$ with center $ O.$ and a line $ L$ which does not touch circle $ C.$ $ OQ$ is perpendicular to $ L,$ $ Q$ is on $ L.$ $ P$ is on $ L,$ draw two tangents $ L_1, L_2$ to circle $ C.$ $ QA, QB$ are perpendicular to $ L_1, L_2$ respectively. ($ A$ on $ L_1,$ $ B$ on $ L_2$). Prove that, line $ AB$ intersect $ QO$ at a fixed point. [i]Original formulation:[/i] A line $ l$ does not meet a circle $ \omega$ with center $ O.$ $ E$ is the point on $ l$ such that $ OE$ is perpendicular to $ l.$ $ M$ is any point on $ l$ other than $ E.$ The tangents from $ M$ to $ \omega$ touch it at $ A$ and $ B.$ $ C$ is the point on $ MA$ such that $ EC$ is perpendicular to $ MA.$ $ D$ is the point on $ MB$ such that $ ED$ is perpendicular to $ MB.$ The line $ CD$ cuts $ OE$ at $ F.$ Prove that the location of $ F$ is independent of that of $ M.$

For a positive constant number $ p$, find $ \lim_{n\to\infty} \frac {1}{n^{p \plus{} 1}}\sum_{k \equal{} 0}^{n \minus{} 1} \int_{2k\pi}^{(2k \plus{} 1)\pi} x^p\sin ^ 3 x\cos ^ 2x\ dx.$