Find the sum of all even positive integers less than $233$ not divisible by $10$.

Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$. Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[ \left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2 \ge \left( \frac{x_1+\dots+x_n}{n} \right) \left( \frac{y_1+\dots+y_n}{n} \right). \] [hide="comment"] [i]Edited by Orl.[/i] [/hide] [i]Proposed by Reid Barton, USA[/i]

Two parallel lines $a$ and $b$ meet two other lines $c$ and $d$. Let $A$ and $A'$ be the points of intersection of $a$ with $c$ and $d$, respectively. Let $B$ and $B'$ be the points of intersection of $b$ with $c$ and $d$, respectively. If $X$ is the midpoint of the line segment $A A'$ and $Y$ is the midpoint of the segment $BB'$, prove that $$|XY| \le \frac{|AB|+|A'B'|}{2}.$$

How many positive solutions are there to $x^{10}+7x^9+14x^8+1729x^7-1379x^6=0$? How many positive integer solutions?

Prove that if $a$ is an integer and $d$ is a positive divisor of the number $a^4+a^3+2a^2-4a+3$, then $d$ is a fourth power modulo $13$.

Let $ABCD$ be a parallelogram and let $E$ be a point on the side $AD$, such that $\frac{AE}{ED} = m$. Let $F$ be a point on $CE$, such that $BF \perp CE$, and the point $G$ is symmetrical to $F$ with respect to $AB$. If point $A$ is the circumcenter of triangle $BFG$, find the value of $m$.

[asy]draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); draw((0,5)--(-2,7)); label("F",(0,0.5),W); label("D",(0,1.5),W); label("C",(0,2.5),W); label("B",(0,3.5),W); label("A",(0,4.5),W); label("A",(0.5,5),N); label("B",(1.5,5),N); label("C",(2.5,5),N); label("D",(3.5,5),N); label("F",(4.5,5),N); label("0",(0.5,0),N); label("0",(0.5,1),N); label("1",(0.5,2),N); label("1",(0.5,3),N); label("2",(0.5,4),N); label("0",(1.5,0),N); label("0",(1.5,1),N); label("3",(1.5,2),N); label("4",(1.5,3),N); label("2",(1.5,4),N); label("2",(2.5,0),N); label("1",(2.5,1),N); label("5",(2.5,2),N); label("3",(2.5,3),N); label("1",(2.5,4),N); label("1",(3.5,0),N); label("1",(3.5,1),N); label("2",(3.5,2),N); label("0",(3.5,3),N); label("0",(3.5,4),N); label("0",(4.5,0),N); label("1",(4.5,1),N); label("0",(4.5,2),N); label("0",(4.5,3),N); label("0",(4.5,4),N); label("TEST 2",(1,6),N); label("TEST 1",(-2,5),SW);[/asy] The table displays the grade distribution of the $ 30$ students in a mathematics class on the last two tests. For example, exactly one student received a "D" on Test 1 and a "C" on Test 2. What percent of the students received the same grade on both tests? \[ \textbf{(A)}\ 12 \% \qquad \textbf{(B)}\ 25 \% \qquad \textbf{(C)}\ 33 \frac{1}{3} \% \qquad \textbf{(D)}\ 40 \% \qquad \textbf{(E)}\ 50 \% \qquad \]

Two circles $\gamma_1$ and $\gamma_2$ cross one another at two points; let $A$ be one of these points. The tangent to $\gamma_1$ at $A$ meets again $\gamma_2$ at $B$, the tangent to $\gamma_2$ at $A$ meets again $\gamma_1$ at $C$, and the line $BC$ meets again $\gamma_1$ and $\gamma_2$ at $D_1$ and $D_2$, respectively. Let $E_1$ and $E_2$ be interior points of the segments $AD_1$ and $AD_2$, respectively, such that $AE_1 = AE_2$. The lines $BE_1$ and $AC$ meet at $M$, the lines $CE_2$ and $AB$ meet at $N$, and the lines $MN$ and $BC$ meet at $P$. Show that the line $PA$ is tangent to the circle $ABC$.

Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.

Let $ABCD$ be a convex quadrilateral with $AD = BC$, $\angle BAC+\angle DCA = 180^{\circ}$, and $\angle BAC \neq 90^{\circ}.$ Let $O_1$ and $O_2$ be the circumcenters of triangles $ABC$ and $CAD$, respectively. Prove that one intersection point of the circumcircles of triangles $O_1BC$ and $O_2AD$ lies on $AC$.

Let $ ABCD$ be a convex quadrilateral. Square $ AB_1A_2B$ is constructed such that the two vertices $ A_2,B_1$ is located outside $ ABCD$. Similarly, we construct squares $ BC_1B_2C$, $ CD_1C_2D$, $ DA_1D_2A$. Let $ K$ be the intersection of $ AA_2$ and $ BB_1$, $ L$ be the intersection of $ BB_2$ and $ CC_1$, $ M$ be the intersection of $ CC_2$ and $ DD_1$, and $ N$ be the intersection of $ DD_2$ and $ AA_1$. Prove that $ KM$ is perpendicular to $ LN$.

Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that - The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, - $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, - The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.

Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$? $\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$

[b]p1.[/b] At a certain point in time, $20\%$ of seniors, $30\%$ of juniors, and $50\%$ of sophomores at a school had a cold. If the number of sick students was the same for each grade, the fraction of sick students across all three grades can be written as $\frac{a}{b}$ , where a and b are relatively prime positive integers. Find $a + b$. [b]p2.[/b] The average score on Mr. Feng’s recent test is a $63$ out of $100$. After two students drop out of the class, the average score of the remaining students on that test is now a $72$. What is the maximum number of students that could initially have been in Mr. Feng’s class? (All of the scores on the test are integers between $0$ and $100$, inclusive.) [b]p3.[/b] Madeline is climbing Celeste Mountain. She starts at $(0, 0)$ on the coordinate plane and wants to reach the summit at $(7, 4)$. Every hour, she moves either $1$ unit up or $1$ unit to the right. A strawberry is located at each of $(1, 1)$ and $(4, 3)$. How many paths can Madeline take so that she encounters exactly one strawberry? [b]p4.[/b] Let $E$ be a point on side $AD$ of rectangle $ABCD$. Given that $AB = 3$, $AE = 4$, and $\angle BEC = \angle CED$, the length of segment $CE$ can be written as $\sqrt{a}$ for some positive integer $a$. Find $a$. [b]p5.[/b] Lucy has some spare change. If she were to convert it into quarters and pennies, the minimum number of coins she would need is $66$. If she were to convert it into dimes and pennies, the minimum number of coins she would need is $147$. How much money, in cents, does Lucy have? [b]p6.[/b] For how many positive integers $x$ does there exist a triangle with altitudes of length $20$, $22$, and $x$? [b]p7.[/b] Compute the number of positive integers $x$ for which $\frac{x^{20}}{x+22}$ is an integer. [b]p8.[/b] Vincent the Bug is crawling along an octagonal prism. He starts on a fixed vertex $A$, visits all other vertices exactly once by traveling along the edges, and returns to $A$. Find the number of paths Vincent could have taken. [b]p9.[/b] Point $U$ is chosen inside square $ALEX$ so that $\angle AUL = 90^o$. Given that $UL = 56$ and $UE = 65$, what is the sum of all possible values for the area of square $ALEX$? [b]p10.[/b] Miranda has prepared $8$ outfits, no two of which are the same quality. She asks her intern Andrea to order these outfits for the new runway show. Andrea first randomly orders the outfits in a list. She then starts removing outfits according to the following method: she chooses a random outfit which is both immediately preceded and immediately succeeded by a better outfit and then removes it. Andrea repeats this process until there are no outfits that can be removed. Given that the expected number of outfits in the final routine can be written as $\frac{a}{b}$ for some relatively prime positive integers $a$ and $b$, find $a + b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way, $6$ small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gray areas is just half the area of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/e/7/a84ad28b3cd45bd0ce455cee2446222fd3eac2.png[/img]

There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that \[ \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.\]