Let $a$, $b$, and $c$ be the roots of the polynomial $$x^3+4x^2-7x-1.$$ Which of the following has roots $ab$, $bc$, and $ac$? $\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.

Let $n$ be an integer greater than $1$. Let $x\in\mathbb R$. (a) Evaluate $S(x,n)=\sum(x+p)(x+q)$, where the summation is over all pairs $(p,q)$ of different numbers from $\{1,2,\ldots,n\}$. (b) Do there exist integers $x,n$ for which $S(x,n)=0$?

Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that \[ \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\] for every positive integer $n$.

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

Positive numbers $x,y$ satisfy equality $$x^3+y^3=x-y$$ Prove that $$x^2+y^2<1$$

If ${a, b}$ and $c$ are positive real numbers, prove that \begin{align*} a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}. \end{align*} [i](Montenegro).[/i]

[b]p4.[/b] What is gcd $(2^6 - 1, 2^9 - 1)$? [b]p5.[/b] Sarah is walking along a sidewalk at a leisurely speed of $\frac12$ m/s. Annie is some distance behind her, walking in the same direction at a faster speed of $s$ m/s. What is the minimum value of $s$ such that Sarah and Annie spend no more than one second within one meter of each other? [b]p6.[/b] You have a choice to play one of two games. In both games, a coin is flipped four times. In game $1$, if (at least) two flips land heads, you win. In game $2$, if (at least) two consecutive flips land heads, you win. Let $N$ be the number of the game that gives you a better chance of winning, and let $p$ be the absolute difference in the probabilities of winning each game. Find $N + p$. PS. You should use hide for answers.

Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\] or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]

In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$? $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$

[b]p1.[/b] If $x$ is a real number that satisfies $\frac{48}{x} = 16$, find the value of $x$. [b]p2.[/b] If $ABC$ is a right triangle with hypotenuse $BC$ such that $\angle ABC = 35^o$, what is $\angle BCA$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/a/b/0f83dc34fb7934281e0e3f988ac34f653cc3f1.png[/img] [b]p3.[/b] If $a\vartriangle b = a + b - ab$, find $4\vartriangle 9$. [b]p4.[/b] Grizzly is $6$ feet tall. He measures his shadow to be $4$ feet long. At the same time, his friend Panda helps him measure the shadow of a nearby lamp post, and it is $6$ feet long. How tall is the lamp post in feet? [b]p5.[/b] Jerry is currently twice as old as Tom was $7$ years ago. Tom is $6$ years younger than Jerry. How many years old is Tom? [b]p6.[/b] Out of the $10, 000$ possible four-digit passcodes on a phone, how many of them contain only prime digits? [b]p7.[/b] It started snowing, which means Moor needs to buy snow shoes for his $6$ cows and $7$ sky bison. A cow has $4$ legs, and a sky bison has $6$ legs. If Moor has 36 snow shoes already, how many more shoes does he need to buy? Assume cows and sky bison wear the same type of shoe and each leg gets one shoe. [b]p8.[/b] How many integers $n$ with $1 \le n \le 100$ have exactly $3$ positive divisors? [b]p9.[/b] James has three $3$ candies and $3$ green candies. $3$ people come in and each randomly take $2$ candies. What is the probability that no one got $2$ candies of the same color? Express your answer as a decimal or a fraction in lowest terms. [b]p10.[/b] When Box flips a strange coin, the coin can land heads, tails, or on the side. It has a $\frac{1}{10}$probability of landing on the side, and the probability of landing heads equals the probability of landing tails. If Box flips a strange coin $3$ times, what is the probability that the number of heads flipped is equal to the number of tails flipped? Express your answer as a decimal or a fraction in lowest terms. [b]p11.[/b] James is travelling on a river. His canoe goes $4$ miles per hour upstream and $6$ miles per hour downstream. He travels $8$ miles upstream and then $8$ miles downstream (to where he started). What is his average speed, in miles per hour? Express your answer as a decimal or a fraction in lowest terms. [b]p12.[/b] Four boxes of cookies and one bag of chips cost exactly $1000$ jelly beans. Five bags of chips and one box of cookies cost less than $1000$ jelly beans. If both chips and cookies cost a whole number of jelly beans, what is the maximum possible cost of a bag of chips? [b]p13.[/b] June is making a pumpkin pie, which takes the shape of a truncated cone, as shown below. The pie tin is $18$ inches wide at the top, $16$ inches wide at the bottom, and $1$ inch high. How many cubic inches of pumpkin filling are needed to fill the pie? [img]https://cdn.artofproblemsolving.com/attachments/7/0/22c38dd6bc42d15ad9352817b25143f0e4729b.png[/img] [b]p14.[/b] For two real numbers $a$ and $b$, let $a\# b = ab - 2a - 2b + 6$. Find a positive real number $x$ such that $(x\#7) \#x = 82$. [b]p15.[/b] Find the sum of all positive integers $n$ such that $\frac{n^2 + 20n + 51}{n^2 + 4n + 3}$ is an integer. [b]p16.[/b] Let $ABC$ be a right triangle with hypotenuse $AB$ such that $AC = 36$ and $BC = 15$. A semicircle is inscribed in $ABC$ as shown, such that the diameter $XC$ of the semicircle lies on side $AC$ and that the semicircle is tangent to $AB$. What is the radius of the semicircle? [img]https://cdn.artofproblemsolving.com/attachments/4/2/714f7dfd09f6da1d61a8f910b5052e60dcd2fb.png[/img] [b]p17.[/b] Let $a$ and $b$ be relatively prime positive integers such that the product $ab$ is equal to the least common multiple of $16500$ and $990$. If $\frac{16500}{a}$ and $\frac{990}{b}$ are both integers, what is the minimum value of $a + b$? [b]p18.[/b] Let $x$ be a positive real number so that $x - \frac{1}{x} = 1$. Compute $x^8 - \frac{1}{x^8}$ . [b]p19.[/b] Six people sit around a round table. Each person rolls a standard $6$-sided die. If no two people sitting next to each other rolled the same number, we will say that the roll is valid. How many dierent rolls are valid? [b]p20.[/b] Given that $\frac{1}{31} = 0.\overline{a_1a_2a_3a_4a_5... a_n}$ (that is, $\frac{1}{31}$ can be written as the repeating decimal expansion $0.a_1a_2... a_na_1a_2... a_na_1a_2...$ ), what is the minimum value of $n$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

Determine all real solutions of the system \begin{align*} x+y+z &=3, \\ \frac1x+\frac1y+\frac1z &= \frac{5}{12}, \\ x^3+y^3+z^3 &=45. \end{align*}

[b]p1.[/b] Consider a normal $8 \times 8$ chessboard, where each square is labelled with either $1$ or $-1$. Let $a_k$ be the product of the numbers in the $k$th row, and let $b_k$ be the product of the numbers in the $k$th column. Find, with proof, all possible values of $\sum^8_{k=1}(a_kb_k)$. [b]p2.[/b] Let $\overline{AB}$ be a line segment with $AB = 1$, and $P$ be a point on $\overline{AB}$ with $AP = x$, for some $0 < x < 1$. Draw circles $C_1$ and $C_2$ with $\overline{AP}$, $\overline{PB}$ as diameters, respectively. Let $\overline{AB_1}$, $\overline{AB_2}$ be tangent to $C_2$ at $B_1$ and $B_2$, and let $\overline{BA_1}$;$\overline{BA_2}$ be tangent to $C_1$ at $A_1$ and $A_2$. Now $C_3$ is a circle tangent to $C_2$, $\overline{AB_1}$, and $\overline{AB_2}$; $C_4$ is a circle tangent to $C_1$, $\overline{BA_1}$, and $\overline{BA_2}$. (a) Express the radius of $C_3$ as a function of $x$. (b) Prove that $C_3$ and $C_4$ are congruent. [img]https://cdn.artofproblemsolving.com/attachments/c/a/fd28ad91ed0a4893608b92f5ccbd01088ae424.png[/img] [b]p3.[/b] Suppose that the graphs of $y = (x + a)^2$ and $x = (y + a)^2$ are tangent to one another at a point on the line $y = x$. Find all possible values of $a$. [b]p4.[/b] You may assume without proof or justification that the infinite radical expressions $\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a-...}}}}$ and $\sqrt{a-\sqrt{a+\sqrt{a-\sqrt{a+...}}}}$ represent unique values for $a > 2$. (a) Find a real number $a$ such that $$\sqrt{a-\sqrt{a-\sqrt{a-\sqrt{a+...}}}}= 2017$$ (b) Show that $$\sqrt{2018-\sqrt{2018+\sqrt{2018-\sqrt{2018+...}}}}=\sqrt{2017-\sqrt{2017-\sqrt{2017-\sqrt{2017-...}}}}$$ [b]p5.[/b] (a) Suppose that $m, n$ are positive integers such that $7n^2 - m^2 > 0$. Prove that, in fact, $7n^2 - m^2 \ge 3$. (b) Suppose that $m, n$ are positive integers such that $\frac{m}{n} <\sqrt7$. Prove that, in fact, $\frac{m}{n}+\frac{1}{mn} <\sqrt7$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that 1. $f$ is a monic polynomial with $\deg f \ge 1$. 2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$. Find all such possible triples. [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

$f_1(x)=x^2+p_1x+q_1,f_2(x)=x^2+p_2x+q_2$ are two parabolas. $l_1$ and $l_2$ are two not parallel lines. It is knows, that segments, that cuted on the $l_1$ by parabolas are equals, and segments, that cuted on the $l_2$ by parabolas are equals too. Prove, that parabolas are equals.

[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible? [img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img] [b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$. [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy? [b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon. [img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img] [b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Represent the polynomial $P(X) = X^{100} + X^{20} + 1$ as the product of 4 polynomials with integer coefficients.

Compute \[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]

Let $a \ge b \ge c > 0$. Prove that $$(a-b+c)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right) \ge 1$$

The sum of the numbers $a,b$ and $c$ is zero, and their product is negative. Prove that the number $\frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}$ is positive.

Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that a) For every integer $n$, we have $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$ b) The value $\frac{1}{2\sqrt{17} }$ is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$