Let $a,b\in\mathbb{R},~a>1,~b>0.$ Find the least possible value for $\alpha$ such that :$$(a+b)^x\geq a^x+b,~(\forall)x\geq\alpha.$$

Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.

Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.

Suppose that $ f: [0,1]\to\mathbb{R}$ has a continuous derivative and that $ \int_0^1f(x)\,dx\equal{}0.$ Prove that for every $ \alpha\in(0,1),$ \[ \left|\int_0^{\alpha}f(x)\,dx\right|\le\frac18\max_{0\le x\le 1}|f'(x)|\]

Find $x\in\mathbb{R}$ for which \[\log_2(x^2 + 4) - \log_2x + x^2 - 4x + 2 = 0.\]

What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Somya has a football game $4$ days from today. If the day before yesterday was Wednesday, what day of the week is the game? [b]p2.[/b] Sammy writes the following equation: $$\frac{2 + 2}{8 + 8}=\frac{x}{8}.$$ What is the value of $x$ in Sammy's equation? [b]p3.[/b] On $\pi$ day, Peter buys $7$ pies. The pies costed $\$3$, $\$1$, $\$4$, $\$1$, $\$5$, $\$9$, and $\$2$. What was the median price of Peter's $7$ pies in dollars? [b]p4.[/b] Antonio draws a line on the coordinate plane. If the line passes through the points ($1, 3$) and ($-1,-1$), what is slope of the line? [b]p5.[/b] Professor Varun has $25$ students in his science class. He divides his students into the maximum possible number of groups of $4$, but $x$ students are left over. What is $x$? [b]p6.[/b] Evaluate the following: $$4 \times 5 \div 6 \times 3 \div \frac47$$ [b]p7.[/b] Jonny, a geometry expert, draws many rectangles with perimeter $16$. What is the area of the largest possible rectangle he can draw? [b]p8.[/b] David always drives at $60$ miles per hour. Today, he begins his trip to MIT by driving $60$ miles. He stops to take a $20$ minute lunch break and then drives for another $30$ miles to reach the campus. What is the total time in minutes he spends getting to MIT? [b]p9.[/b] Richard has $5$ hats: blue, green, orange, red, and purple. Richard also has 5 shirts of the same colors: blue, green, orange, red, and purple. If Richard needs a shirt and a hat of different colors, how many outts can he wear? [b]p10.[/b] Poonam has $9$ numbers in her bag: $1, 2, 3, 4, 5, 6, 7, 8, 9$. Eric runs by with the number $36$. How many of Poonam's numbers evenly divide Eric's number? [b]p11.[/b] Serena drives at $45$ miles per hour. If her car runs at $6$ miles per gallon, and each gallon of gas costs $2$ dollars, how many dollars does she spend on gas for a $135$ mile trip? [b]p12.[/b] Grace is thinking of two integers. Emmie observes that the sum of the two numbers is $56$ but the difference of the two numbers is $30$. What is the sum of the squares of Grace's two numbers? [b]p13.[/b] Chang stands at the point ($3,-3$). Fang stands at ($-3, 3$). Wang stands in-between Chang and Fang; Wang is twice as close to Fang as to Chang. What is the ordered pair that Wang stands at? [b]p14.[/b] Nithin has a right triangle. The longest side has length $37$ inches. If one of the shorter sides has length $12$ inches, what is the perimeter of the triangle in inches? [b]p15.[/b] Dora has $2$ red socks, $2$ blue socks, $2$ green socks, $2$ purple socks, $3$ black socks, and $4$ gray socks. After a long snowstorm, her family loses electricity. She picks socks one-by-one from the drawer in the dark. How many socks does she have to pick to guarantee a pair of socks that are the same color? [b]p16.[/b] Justin selects a random positive $2$-digit integer. What is the probability that the sum of the two digits of Justin's number equals $11$? [b]p17.[/b] Eddie correctly computes $1! + 2! + .. + 9! + 10!$. What is the remainder when Eddie's sum is divided by $80$? [b]p18.[/b] $\vartriangle PQR$ is drawn such that the distance from $P$ to $\overline{QR}$ is $3$, the distance from $Q$ to $\overline{PR}$ is $4$, and the distance from $R$ to $\overline{PQ}$ is $5$. The angle bisector of $\angle PQR$ and the angle bisector of $\angle PRQ$ intersect at $I$. What is the distance from $I$ to $\overline{PR}$? [b]p19.[/b] Maxwell graphs the quadrilateral $|x - 2| + |y + 2| = 6$. What is the area of the quadrilateral? [b]p20.[/b] Uncle Gowri hits a speed bump on his way to the hospital. At the hospital, patients who get a rare disease are given the option to choose treatment $A$ or treatment $B$. Treatment $A$ will cure the disease $\frac34$ of the time, but since the treatment is more expensive, only $\frac{8}{25}$ of the patients will choose this treatment. Treatment $B$ will only cure the disease $\frac{1}{2}$ of the time, but since it is much more aordable, $\frac{17}{25}$ of the patients will end up selecting this treatment. Given that a patient was cured, what is the probability that the patient selected treatment $A$? [b]p21.[/b] In convex quadrilateral $ABCD$, $AC = 28$ and $BD = 15$. Let $P, Q, R, S$ be the midpoints of $AB$, $BC$, $CD$ and $AD$ respectively. Compute $PR^2 + QS^2$. [b]p22.[/b] Charlotte writes the polynomial $p(x) = x^{24} - 6x + 5$. Let its roots be $r_1$, $r_2$, $...$, $r_{24}$. Compute $r^{24}_1 +r^{24}_2 + r^{24}_3 + ... + r^{24}_24$. [b]p23.[/b] In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Let $E$ be a point on $CD$, and let $F$ be the point on $AB$ which lies on the bisector of $\angle BED$. If $FD^2 + EF^2 = 52$, what is the length of $BE$? [b]p24.[/b] In $\vartriangle ABC$, the measure of $\angle A$ is $60^o$ and the measure of $\angle B$ is $45^o$. Let $O$ be the center of the circle that circumscribes $\vartriangle ABC$. Let $I$ be the center of the circle that is inscribed in $\vartriangle ABC$. Finally, let $H$ be the intersection of the $3$ altitudes of the triangle. What is the angle measure of $\angle OIH$ in degrees? [b]p25.[/b] Kaitlyn fully expands the polynomial $(x^2 + x + 1)^{2018}$. How many of the coecients are not divisible by $3$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality : $\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$

For positive numbers $ a_1,a_2,\dots,a_n$, we define \[ A\equal{}\frac{a_1\plus{}a_2\plus{}\cdots\plus{}a_n}{n}, \quad G\equal{}\sqrt[n]{a_1\cdots a_n}, \quad H\equal{}\frac{n}{a_1^{\minus{}1}\plus{}\cdots\plus{}a_n^{\minus{}1}}\] Prove that (i) $ \frac{A}{H}\leq \minus{}1\plus{}2\left(\frac{A}{G}\right)^n$, for n even (ii) $ \frac{A}{H}\leq \minus{}\frac{n\minus{}2}{n}\plus{}\frac{2(n\minus{}1)}{n}\left(\frac{A}{G}\right)^n$, for $ n$ odd

i) Prove that for every sequence $(a_{n})_{n\in \mathbb{N}}$, such that $a_{n}>0$ for all $n \in \mathbb{N}$ and $\sum_{n=1}^{\infty}a_{n}<\infty$, we have $$\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}.$$ ii) Prove that for every $\epsilon>0$ there exists a sequence $(b_{n})_{n\in \mathbb{N}}$ such that $b_{n}>0$ for all $n \in \mathbb{N}$ and $\sum_{n=1}^{\infty}b_{n}<\infty$ and $$\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.$$

a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1[/b] What is the sum of the first $5$ positive integers? [b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$? [b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number? [b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen? [b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$? [b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle? [b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus? [b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win? [b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece? [b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$? [b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime? Note: $1$ is not prime. [b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement? [b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares? [b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions? [b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card? [b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$? [b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$? [b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$? [b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$. [b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$. [b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once. [b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle. [b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself. [b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe? [b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$? [b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$. [b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers. [b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days. [b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it. [b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$. [b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places. [b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect. [b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$. [b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ . [b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$. [b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$? [b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$. [b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$. [b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ . [b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$. [b]p23.[/b] Let $a,b,c$ be positive real numbers such that (i) $c > a$ (ii) $10c = 7a +4b +2024$ (iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$. Find $a +b +c$. [b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$. [b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.

[b]a)[/b] Let $ a,b $ two non-negative integers such that $ a^2>b. $ Show that the equation $$ \left\lfloor\sqrt{x^2+2ax+b}\right\rfloor =x+a-1 $$ has an infinite number of solutions in the non-negative integers. Here, $ \lfloor\alpha\rfloor $ denotes the floor of $ \alpha. $ [b]b)[/b] Find the floor of $ m=\sqrt{2+\sqrt{2+\underbrace{\cdots}_{\text{n times}}+\sqrt{2}}} , $ where $ n $ is a natural number. Justify.