[b]4.[/b] Let $\left (H_{\alpha} \right ) $ be a system of sets of integers having the property that for any $\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}$ is a finite set and $H_{{\alpha} _1} \neq H_{{\alpha} _2}$. Prove that there exists a system $\left (H_{\alpha} \right )$ of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets $H_\alpha$ contains more than $K$ elements, then the system $\left (H_{\alpha} \right ) $ is countable ($K$ is an arbitrary fixed integer). [b](St. 4)[/b]

Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior? $ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $

On the coordinate plane, draw all points$ M(x, y)$, the coordinates of which satisfy the inequalities $$\cos(x + y)^2 \le \cos(x - y)^2, \,\,\, 0 \le x^3, \,\,\, 0 \le y^3.$$

Find, with proof, all triples of positive integers $(x,y,z)$ satisfying the equation $3^x - 5^y = 4z^2$.

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically? $\textbf{(A)}$ If Lewis did not receive an A, then he got all of the multiple choice questions wrong. \\ $\textbf{(B)}$ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. \\ $\textbf{(C)}$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. \\ $\textbf{(D)}$ If Lewis received an A, then he got all of the multiple choice questions right. \\ $\textbf{(E)}$ If Lewis received an A, then he got at least one of the multiple choice questions right.

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

Let $S$ denote a square of the side-length $7$, and let eight squares of the side-length $3$ be given. Show that $S$ can be covered by those eight small squares.

Show that there are infinitely many tuples $(a,b,c,d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ($A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard). [i]V. Chukanov[/i]

Determine the smallest integer $n \ge 1$ such that a $n \times n$ square can be cut into square pieces of size $1 \times 1$ and $2 \times 2$ with both types occuring the same number of times.

Side $AB$ of triangle $ABC$ has length $8$ inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: $\textbf{(A)}\ \dfrac{51}{4} \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ \dfrac{53}{4} \qquad \textbf{(D)}\ \dfrac{40}{3} \qquad \textbf{(E)}\ \dfrac{27}{2} $

Given a positive integer $n$. Prove the inequality $\sum\limits_{i=1}^{n}\frac{1}{i(i+1)(i+2)(i+3)(i+4)}<\frac{1}{96}$

What is the sum $ \frac{1}{1 \cdot 2 \cdot 3} \plus{} \frac{1}{2 \cdot 3 \cdot 4} \plus{} \cdots \plus{} \frac{1}{1996 \cdot 1997 \cdot 1998}$? A. $ \frac{2 \cdot 1997}{3 \cdot 1996 \cdot 1998}$ B. $ \frac{1}{3} \minus{} \frac{1}{3 \cdot 1998}$ C. $ \frac{1}{4} \minus{} \frac{1}{1997^2}$ D. $ \frac{1}{3} \minus{} \frac{1}{3 \cdot 1997 \cdot 1998}$ E. $ \frac{1}{4} \minus{} \frac{1}{2 \cdot 1997 \cdot 1998}$

Let $n$ be a positive integer, and let $f(n) =1^n + 2^{n-1} + 3^{n-2}+ 4^{n-3}+... + (n-1)^2 + n^1$ Find the smallest possible value of $\frac{f(n+2)}{f(n)}$ .Justify your answer.

A man drives $ 150$ miles to the seashore in $ 3$ hours and $ 20$ minutes. He returns from the shore to the starting point in $ 4$ hours and $ 10$ minutes. Let $ r$ be the average rate for the entire trip. Then the average rate for the trip going exceeds $ r$ in miles per hour, by: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \frac{1}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$

Mr. DoBa likes to listen to music occasionally while he does his math homework. When he listens to classical music, he solves one problem every $3$ minutes. When he listens to rap music, however, he only solves one problem every $5$ minutes. Mr. DoBa listens to a playlist comprised of $60\%$ classical music and $40\%$ rap music. Each song is exactly $4$ minutes long. Suppose that the expected number of problems he solves in an hour does not depend on whether or not Mr. DoBa is listening to music at any given moment, and let $m$ the average number of problems Mr. DoBa solves per minute when he is not listening to music. Determine the value of $1000m$.

Lines from the vertices of a unit square are drawn to the midpoints of the sides as shown in the figure below. What is the area of quadrilateral $ABCD$? Express your answer in simplest terms. [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,0)--(1,0.5)); draw((1,0)--(0.5,1)); draw((1,1)--(0,0.5)); draw((0,1)--(0.5,0)); label("$A$",(0.2,0.6),N); label("$B$",(0.4,0.2),W); label("$C$",(0.8,0.4),S); label("$D$",(0.6,0.8),E); [/asy] $\text{(A) }\frac{\sqrt{2}}{9}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{\sqrt{3}}{9}\qquad\text{(D) }\frac{\sqrt{8}}{8}\qquad\text{(E) }\frac{1}{5}$

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] What is $20 \div 2 - 0 \times 1 + 2 \times 5$? [b]p2.[/b] Today is Saturday, January $25$, $2020$. Exactly four hundred years from today, January $25$, $2420$, is again a Saturday. How many weekend days (Saturdays and Sundays) are in February, $2420$? (January has $31$ days and in year $2040$, February has $29$ days.) [b]p3.[/b] Given that there are four people sitting around a circular table, and two of them stand up, what is the probability that the two of them were originally sitting next to each other? [b]p4.[/b] What is the area of a triangle with side lengths $5$, $5$, and $6$? [b]p5.[/b] Six people go to OBA Noodles on Main Street. Each person has $1/2$ probability to order Duck Noodle Soup, $1/3$ probability to order OBA Ramen, and $1/6$ probability to order Kimchi Udon Soup. What is the probability that three people get Duck Noodle Soup, two people get OBA Ramen, and one person gets Kimchi Udon Soup? [b]p6.[/b] Among all positive integers $a$ and $b$ that satisfy $a^b = 64$, what is the minimum possible value of $a+b$? [b]p7.[/b] A positive integer $n$ is called trivial if its tens digit divides $n$. How many two-digit trivial numbers are there? [b]p8.[/b] Triangle $ABC$ has $AB = 5$, $BC = 13$, and $AC = 12$. Square $BCDE$ is constructed outside of the triangle. The perpendicular line from $A$ to side $DE$ cuts the square into two parts. What is the positive difference in their areas? [b]p9.[/b] In an increasing arithmetic sequence, the first, third, and ninth terms form an increasing geometric sequence (in that order). Given that the first term is $5$, find the sum of the first nine terms of the arithmetic sequence. [b]p10.[/b] Square $ABCD$ has side length $1$. Let points $C'$ and $D'$ be the reflections of points $C$ and $D$ over lines $AB$ and $BC$, respectively. Let P be the center of square $ABCD$. What is the area of the concave quadrilateral $PD'BC'$? [b]p11.[/b] How many four-digit palindromes are multiples of $7$? (A palindrome is a number which reads the same forwards and backwards.) [b]p12.[/b] Let $A$ and $B$ be positive integers such that the absolute value of the difference between the sum of the digits of $A$ and the sum of the digits of $(A + B)$ is $14$. What is the minimum possible value for $B$? [b]p13.[/b] Clark writes the following set of congruences: $x \equiv a$ (mod $6$), $x \equiv b$ (mod $10$), $x \equiv c$ (mod $15$), and he picks $a$, $b$, and $c$ to be three randomly chosen integers. What is the probability that a solution for $x$ exists? [b]p14.[/b] Vincent the bug is crawling on the real number line starting from $2020$. Each second, he may crawl from $x$ to $x - 1$, or teleport from $x$ to $\frac{x}{3}$ . What is the least number of seconds needed for Vincent to get to $0$? [b]p15.[/b] How many positive divisors of $2020$ do not also divide $1010$? [b]p16.[/b] A bishop is a piece in the game of chess that can move in any direction along a diagonal on which it stands. Two bishops attack each other if the two bishops lie on the same diagonal of a chessboard. Find the maximum number of bishops that can be placed on an $8\times 8$ chessboard such that no two bishops attack each other. [b]p17.[/b] Let $ABC$ be a right triangle with hypotenuse $20$ and perimeter $41$. What is the area of $ABC$? [b]p18.[/b] What is the remainder when $x^{19} + 2x^{18} + 3x^{17} +...+ 20$ is divided by $x^2 + 1$? [b]p19.[/b] Ben splits the integers from $1$ to $1000$ into $50$ groups of $20$ consecutive integers each, starting with $\{1, 2,...,20\}$. How many of these groups contain at least one perfect square? [b]p20.[/b] Trapezoid $ABCD$ with $AB$ parallel to $CD$ has $AB = 10$, $BC = 20$, $CD = 35$, and $AD = 15$. Let $AD$ and $BC$ intersect at $P$ and let $AC$ and $BD$ intersect at $Q$. Line $PQ$ intersects $AB$ at $R$. What is the length of $AR$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a triangle $ABC$ and a point $P$ on the side $AB$ . Let $Q$ be the intersection of the straight line $CP$ (different from $C$) with the circumcicle of the triangle. Prove the inequality $$\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2$$ and that equality holds if and only if the $CP$ is bisector of the angle $ACB$. [img]https://cdn.artofproblemsolving.com/attachments/b/1/068fafd5564e77930160115a1cd409c4fdbf61.png[/img]

Let $n$ be any positive integer, and let $S(n)$ denote the number of permutations $\tau$ of $\{1,\dots,n\}$ such that $k^4+(\tau(k))^4$ is prime for all $k=1,\dots,n$. Show that $S(n)$ is always a square.

Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that \[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]

The positive integer equal to the expression \[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\] is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors. [i]Team #7[/i]

Let point $I$ be the center of the inscribed circle of an acute-angled triangle $ABC$. Rays $AI$ and $BI$ intersect the circumcircle $k$ of triangle $ABC$ at points $D$ and $E$ respectively. The segments $DE$ and $CA$ intersect at point $F$, the line through point $E$ parallel to the line $FI$ intersects the circle $k$ at point $G$, and the lines $FI$ and $DG$ intersect at point $H$. Prove that the lines $CA$ and $BH$ touch the circumcircle of the triangle $DFH$ at the points $F$ and $H$ respectively.

The external bisectors of the angles of the convex quadrilateral $ABCD$ intersect each other in $E,F,G$ and $H$ such that $A\in EH, \ B\in EF, \ C\in FG, \ D\in GH$. We know that the perpendiculars from $E$ to $AB$, from $F$ to $BC$ and from $G$ to $CD$ are concurrent. Prove that $ABCD$ is cyclic.

Let $\, D \,$ be an arbitrary point on side $\, AB \,$ of a given triangle $\, ABC, \,$ and let $\, E \,$ be the interior point where $\, CD \,$ intersects the external common tangent to the incircles of triangles $\, ACD \,$ and $\, BCD$. As $\, D \,$ assumes all positions between $\, A \,$ and $\, B \,$, prove that the point $\, E \,$ traces the arc of a circle.