Let $P$ be a point inside a triangle $ABC$ such that $\angle PAC= \angle PCB$. Let the projections of $P$ onto $BC$, $CA$, and $AB$ be $X,Y,Z$ respectively. Let $O$ be the circumcenter of $\triangle XYZ$, $H$ be the foot of the altitude from $B$ to $AC$, $N$ be the midpoint of $AC$, and $T$ be the point such that $TYPO$ is a parallelogram. Show that $\triangle THN$ is similar to $\triangle PBC$. [i]Proposed by Sammy Luo[/i]

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

Let $k$ be a given positive integer. The sequence $x_n$ is defined as follows: $x_1 =1$ and $x_{n+1}$ is the least positive integer which is not in $\{x_{1}, x_{2},..., x_{n}, x_{1}+k, x_{2}+2k,..., x_{n}+nk \}$. Show that there exist real number $a$ such that $x_n = \lfloor an\rfloor$ for all positive integer $n$.

Let $T$ be the answer to question $17$, and let $N =\frac{24}{T}$. Leanne flips a fair coin $N$ times. Let $X$ be the number of times that within a series of three consecutive flips, there were exactly two heads or two tails. What is the expected value of $X$?

The function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called [i]“Sozopolian”[/i], if it satisfies the following two properties: For each two $x,y\in \mathbb{Z}$ which aren’t multiples of 17 the number $f(xy)-f(x)-f(y)$ is divisible by 8; For $\forall x\in \mathbb{Z}$ the number $f(x+17)-f(x)$ is divisible by 8. Does there exist a [i]Sozopolian[/i] function for which a) $f(2)=1; \quad$ b) $f(3)=1$?

Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$

Let $ Z_1,\,Z_2\dots,\,Z_n$ be $ d$-dimensional independent random (column) vectors with standard normal distribution, $ n \minus{} 1 > d$. Furthermore let \[ \overline Z \equal{} \frac {1}{n}\sum_{i \equal{} 1}^n Z_i,\quad S_n \equal{} \frac {1}{n \minus{} 1}\sum_{i \equal{} 1}^n(Z_i \minus{} \overline Z)(Z_i \minus{} \overline Z)^\top\] be the sample mean and corrected empirical covariance matrix. Consider the standardized samples $ Y_i \equal{} S_n^{ \minus{} 1/2}(Z_i \minus{} \overline Z)$, $ i \equal{} 1,2,\dots,n$. Show that \[ \frac {E|Y_1 \minus{} Y_2|}{E|Z_1 \minus{} Z_2|} > 1,\] and that the ratio does not depend on $ d$, only on $ n$.

Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying $$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$

Let $ABC$ be a triangle. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AD}$, and let $F$ be the midpoint of $\overline{BE}$. Let $G$ be the point where the lines $AB$ and $CF$ intersect. What is the value of $\frac{AG}{AB}$?

We have seven equal pails with water, filled to one half, one third, one quarter, one fifth, one eighth, one ninth, and one tenth, respectively. We are allowed to pour water from one pail into another until the first pail empties or the second one fills to the brim. Can we obtain a pail that is filled to (a) one twelfth, (b) one sixth after several such steps?

[b]p1.[/b] David is taking a $50$-question test, and he needs to answer at least $70\%$ of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test? [b]p2.[/b] You decide to flip a coin some number of times, and record each of the results. You stop flipping the coin once you have recorded either $20$ heads, or $16$ tails. What is the maximum number of times that you could have flipped the coin? [b]p3.[/b] The width of a rectangle is half of its length. Its area is $98$ square meters. What is the length of the rectangle, in meters? [b]p4.[/b] Carol is twice as old as her younger brother, and Carol's mother is $4$ times as old as Carol is. The total age of all three of them is $55$. How old is Carol's mother? [b]p5.[/b] What is the sum of all two-digit multiples of $9$? [b]p6.[/b] The number $2016$ is divisible by its last two digits, meaning that $2016$ is divisible by $16$. What is the smallest integer larger than $2016$ that is also divisible by its last two digits? [b]p7.[/b] Let $Q$ and $R$ both be squares whose perimeters add to $80$. The area of $Q$ to the area of $R$ is in a ratio of $16 : 1$. Find the side length of $Q$. [b]p8.[/b] How many $8$-digit positive integers have the property that the digits are strictly increasing from left to right? For instance, $12356789$ is an example of such a number, while $12337889$ is not. [b]p9.[/b] During a game, Steve Korry attempts $20$ free throws, making 16 of them. How many more free throws does he have to attempt to finish the game with $84\%$ accuracy, assuming he makes them all? [b]p10.[/b] How many dierent ways are there to arrange the letters $MILKTEA$ such that $TEA$ is a contiguous substring? For reference, the term "contiguous substring" means that the letters $TEA$ appear in that order, all next to one another. For example, $MITEALK$ would be such a string, while $TMIELKA$ would not be. [b]p11.[/b] Suppose you roll two fair $20$-sided dice. What is the probability that their sum is divisible by $10$? [b]p12.[/b] Suppose that two of the three sides of an acute triangle have lengths $20$ and $16$, respectively. How many possible integer values are there for the length of the third side? [b]p13.[/b] Suppose that between Beijing and Shanghai, an airplane travels $500$ miles per hour, while a train travels at $300$ miles per hour. You must leave for the airport $2$ hours before your flight, and must leave for the train station $30$ minutes before your train. Suppose that the two methods of transportation will take the same amount of time in total. What is the distance, in miles, between the two cities? [b]p14.[/b] How many nondegenerate triangles (triangles where the three vertices are not collinear) with integer side lengths have a perimeter of $16$? Two triangles are considered distinct if they are not congruent. [b]p15.[/b] John can drive $100$ miles per hour on a paved road and $30$ miles per hour on a gravel road. If it takes John $100$ minutes to drive a road that is $100$ miles long, what fraction of the time does John spend on the paved road? [b]p16.[/b] Alice rolls one pair of $6$-sided dice, and Bob rolls another pair of $6$-sided dice. What is the probability that at least one of Alice's dice shows the same number as at least one of Bob's dice? [b]p17.[/b] When $20^{16}$ is divided by $16^{20}$ and expressed in decimal form, what is the number of digits to the right of the decimal point? Trailing zeroes should not be included. [b]p18.[/b] Suppose you have a $20 \times 16$ bar of chocolate squares. You want to break the bar into smaller chunks, so that after some sequence of breaks, no piece has an area of more than $5$. What is the minimum possible number of times that you must break the bar? For an example of how breaking the chocolate works, suppose we have a $2\times 2$ bar and wish to break it entirely into $1\times 1$ bars. We can break it once to get two $2\times 1$ bars. Then, we would have to break each of these individual bars in half in order to get all the bars to be size $1\times 1$, and we end up using $3$ breaks in total. [b]p19.[/b] A class of $10$ students decides to form two distinguishable committees, each with $3$ students. In how many ways can they do this, if the two committees can have no more than one student in common? [b]p20.[/b] You have been told that you are allowed to draw a convex polygon in the Cartesian plane, with the requirements that each of the vertices has integer coordinates whose values range from $0$ to $10$ inclusive, and that no pair of vertices can share the same $x$ or $y$ coordinate value (so for example, you could not use both $(1, 2)$ and $(1, 4)$ in your polygon, but $(1, 2)$ and $(2, 1)$ is fine). What is the largest possible area that your polygon can have? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Geri plays chess against himself. White has a 5% chance of winning, Black has a 5% chance of winning, and there is a 90% chance of a draw. What is the expected number of games Geri will have to play against himself for one of the colors to win four times? [i]Proposed by Matthew Weiss

Let $ABC$ be a triangle. Let $D$ be the intersection point of the angle bisector at $A$ with $BC$. Let $T$ be the intersection point of the tangent line to the circumcircle of triangle $ABC$ at point $A$ with the line through $B$ and $C$. Let $I$ be the intersection point of the orthogonal line to $AT$ through point $D$ with the altitude $h_a$ of the triangle at point $A$. Let $P$ be the midpoint of $AB$, and let $O$ be the circumcenter of triangle $ABC$. Let $M$ be the intersection point of $AB$ and $TI$, and let $F$ be the intersection point of $PT$ and $AD$. Prove: $MF$ and $AO$ are orthogonal to each other.

If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula \[ \binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, . \] Compute the value of \[ \frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, . \]