Several napkins of equal size and of shape of a unit disc were placed on a table (with overlappings). Is it always possible to hammer several point-sized nails so that all the napkins will be thus attached to the table with the same number of nails? (The nails cannot be hammered into the borders of the discs). Vladimir Dolnikov, Pavel Kozhevnikov

The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers? $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$

A positive integer $n$ is called $omopeiro$ if there exists $n$ non-zero integers that are not necessarily distinct such that $2021$ is the sum of the squares of those $n$ integers. For example, the number $2$ is not an $omopeiro$, because $2021$ is not a sum of two non-zero squares, but $2021$ is an $omopeiro$, because $2021=1^2+1^2+ \dots +1^2$, which is a sum of $2021$ squares of the number $1$. Prove that there exist more than 1500 $omopeiro$ numbers. Note: proving that there exist at least 500 $omopeiro$ numbers is worth 2 points.

Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.

Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$, respectively. Show that there is a point common to $\Omega$, the circumcircle of $DXY$, and the line through $A$ perpendicular to $CE$. [i]Proposed by Michael Ren and Vincent Huang[/i]

Find all the quaterns $(x,y,z,w)$ of real numbers (not necessarily distinct) that solve the following system of equations: $$x+y=z^2+w^2+6zw$$ $$x+z=y^2+w^2+6yw$$ $$x+w=y^2+z^2+6yz$$ $$y+z=x^2+w^2+6xw$$ $$y+w=x^2+z^2+6xz$$ $$z+w=x^2+y^2+6xy$$

Each of two angles of a triangle is $60^{\circ}$ and the included side is $4$ inches. The area of the triangle, in square inches, is: $ \textbf{(A) }8\sqrt{3}\qquad\textbf{(B) }8\qquad\textbf{(C) }4\sqrt{3}\qquad\textbf{(D) }4\qquad\textbf{(E) }2\sqrt{3} $

[b]1.[/b] Solve without use of determinants the following system of linear equations: $\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ($k= 0,1, \dots , n$), where $\alpha$ is a fixed real number. [b](A. 7)[/b]

[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].

Adamand Topher are playing a game in which each of them starts with $2$ pickles. Each turn, they flip a fair coin: if it lands heads, Topher takes $1$ pickle from Adam; if it lands tails, Adam takes $2$ pickles from Topher. (If Topher has only $1$ pickle left, Adam will just take it.) What’s the probability that Topher will have all $4$ pickles before Adam does?

Pentagon $SPEAK$ is inscribed in triangle $NOW$ such that $S$ and $P$ lie on segment $NO$, $K$ and $A$ lie on segment $NW$, and $E$ lies on segment $OW$. Suppose that $NS = SP = PO$ and $NK = KA = AW$. Given that $EP = EK = 5$ and $EA = ES = 6$, compute $OW$.

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

Consider a triangle $ABC$, with the points $A_1$, $A_2$ on side $BC$, $B_1,B_2\in\overline{AC}$, $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$, $BA_1<BA_2$, $CB_1<CB_2$. Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$. Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$, let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$. Prove that the lines $AA^*$, $BB^*$, $CC^*$ are concurrent.

Let $ABC$ be an isosceles obtuse-angled triangle, and $D$ be a point on its base $AB$ such that $AD$ equals to the circumradius of triangle $BCD$. Find the value of $\angle ACD$.

Given $n\times n$ square table, fill in each square with $-1$ or $1$. For $n$ squares that any two of them are neither in the same line nor in the same column, we call the product of the numbers in the squares [i]basic term[/i]. Prove that she sum of all [i]basic terms[/i] is always a multiple of $4$.

Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$. [i]Laurentiu Panaitopol[/i]

Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$. Find the product $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.

Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5. If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$, then the slip with 3.5 must go into what cup? $ \textbf{(A) } A \qquad \textbf{(B) } B \qquad \textbf{(C) } C \qquad \textbf{(D) } D \qquad \textbf{(E) } E $

The circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ are externally tangent at the point $T$. The circle $\omega_3$ centered at $O_3$ is tangent to the line $AB$ (the external common tangent of $\omega_1$ and $\omega_2$) at $D$ and externally tangent to $\omega_1$ and to $\omega_2$. The line $TD$ intersects again at $\omega_1$. Prove that $O_1 C \parallel AB$. [I]Proposed by V. Rastorguev[/I]

The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r, s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $ t \ne 0$. Compute $|f(0)|$.

Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$ for any complex number $x.$

For every subset $\mathcal{P}$ of the plane let $S(\mathcal{P})$ denote the set of circles and lines that intersect $\mathcal{P}$ in at least three points. Find all sets $\mathcal{P}$ consisting of 2024 points such that for any two distinct elements of $S(\mathcal{P}),$ their intersection points all belong to $\mathcal{P}{}.$

Let $n$ be a positive integer. There are $n$ purple and $n$ white cows queuing in a line in some order. Tim wishes to sort the cows by colour, such that all purple cows are at the front of the line. At each step, he is only allowed to swap two adjacent groups of equally many consecutive cows. What is the minimal number of steps Tim needs to be able to fulfill his wish, regardless of the initial alignment of the cows?

The circles $ S_1 $ and $ S_2 $ intersect at points $ A $ and $ B $. Circle $ S_3 $ externally touches $ S_1 $ and $ S_2 $ at points $ C $ and $ D $ respectively. Let $ K $ be the midpoint of the chord cut by the line $ AB $ on circles $ S_3 $. Prove that $ \angle CKA = \angle DKA $.