In the grid shown, fill in each empty space with a number, such that after the grid is completely filled in, the number in each space is equal to the smallest positive integer that does not appear in any of the touching spaces. (A pair of spaces is considered to touch if they both share a vertex.) You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(.5cm); int count=0; real s3=1.73205081; pair[] points={(-3,3*s3), (-1,3*s3), (1,3*s3), (3,3*s3), (-4,2*s3), (-2,2*s3), (0,2*s3), (2,2*s3), (4,2*s3), (-5,s3), (-3,s3), (-1,s3), (1,s3), (3,s3), (5,s3), (-6,0), (-4,0), (-2,0), (0,0), (2,0), (4,0), (6,0), (-5,-s3), (-3,-s3), (-1,-s3), (1,-s3), (3,-s3), (5,-s3), (-4,-2*s3), (-2,-2*s3), (0,-2*s3), (2,-2*s3), (4,-2*s3), (-3,-3*s3), (-1,-3*s3), (1,-3*s3), (3,-3*s3)}; void draw_hexagon(pair p) { draw(shift(p)*scale(2/s3)*(dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--dir(30))); } void add_number() { draw_hexagon(points[count]); count+=1; } void add_number(int n) { label((string)n, points[count]); add_number(); } add_number(4); add_number(); add_number(); add_number(1); add_number(1); add_number(); add_number(1); add_number(); add_number(3); add_number(); add_number(2); add_number(); add_number(); add_number(6); add_number(); add_number(); add_number(); add_number(); add_number(7); add_number(); add_number(); add_number(); add_number(); add_number(3); add_number(); add_number(); add_number(5); add_number(); add_number(2); add_number(); add_number(4); add_number(); add_number(3); add_number(3); add_number(); add_number(); add_number(2);[/asy]

All empty white triangles in figure are to be filled with integers such that for each gray triangle the three numbers in the white neighboring triangles sum to a multiple of $5$. The lower left and the lower right white triangle are already filled with the numbers $12$ and $3$, respectively. Find all integers that can occur in the uppermost white triangle. (G. Woeginger, Eindhoven, The Netherlands) [img]https://cdn.artofproblemsolving.com/attachments/8/a/764732f5debbd58a147e9067e83ba8d31f7ee9.png[/img]

What is the maximum number of non-overlapping $ 2\times 1$ dominoes that can be placed on a $ 8\times 9$ checkerboard if six of them are placed as shown? Each domino must be placed horizontally or vertically so as to cover two adjacent squares of the board.

Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?

The numbers $1, 2, 3, ..., 1000$ are written on the board. Two people take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by three, then the one who made the first move wins. if not, then his partner. Which one will win if played correctly?

In circle $\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_1$ and $M_2$ respectively, such that $PM_1=15$ and $PM_2=20$. Line $M_1M_2$ intersects $\omega$ at points $A$ and $B$, with $M_1$ between $A$ and $M_2$. Compute the largest possible value of $BM_2-AM_1$.

Suppose that $F,G,H$ are polynomials of degree at most $2n+1$ with real coefficients such that: i) For all real $x$ we have $F(x)\le G(x)\le H(x)$. ii) There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)\quad\text{for}\ i=1,2,3,\ldots ,n$. iii) There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$. Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.

Let $ABCD$ be a quadrilateral inscribed in circle $\omega$. Define $E = AA \cap CD$, $F = AA \cap BC$, $G = BE \cap \omega$, $H = BE \cap AD$, $I = DF \cap \omega$, and $J = DF \cap AB$. Prove that $GI$, $HJ$, and the $B$-symmedian are concurrent. [i]Proposed by Robin Park[/i]

Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$. (b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$. (c) $|A| = |B| = |C| = 4$. Here, $|S|$ denotes the number of elements in the set $S$.

How many $10$-digit sequences are there, made up of $1$ four, $2$ threes, $3$ twos, and $4$ ones, in which there is a two in between any two ones, a three in between any two twos, and a four in between any two threes?

The math team at Jupiter Falls Middle School meets twice a month during the Summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations: \begin{align*}2x+3y+3z&=8,\\3x+2y+3z&=808,\\3x+3y+2z&=80808.\end{align*} Their goal is not to find a solution $(x,y,z)$ to the system, but instead to compute the sum of the variables. Find the value of $x+y+z$.

Let $ABC$ be a triangle with $I$ as incenter.The incircle touches $BC$ at $D$.Let $D'$ be the antipode of $D$ on the incircle.Make a tangent at $D'$ to incircle.Let it meet $(ABC)$ at $X,Y$ respectively.Let the other tangent from $X$ meet the other tangent from $Y$ at $Z$.Prove that $(ZBD)$ meets $IB$ at the midpoint of $IB$

Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.

A convex quadrilateral $ABCD$ is given where $\angle DAB =\angle ABC = 120^o$ and $CD = 3$,$BC = 2$, $AB = 1$. Calculate the length of segment $AD$.

$$\int_{-\infty}^\infty \frac{\sin(\pi x)}{x(1+x^2)}\,dx$$ [i]Proposed by Vlad Oleksenko[/i]

Let $n$ be a positive integer and let vectors $v_1$, $v_2$, $\ldots$, $v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i$th minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n$th minute there exists no point which is occupied by the flea with greater probability than the origin. [i]Proposed by Péter Pál Pach, Budapest[/i]

Compute \[\sum_{n_{60}=0}^2\sum_{n_{59}=0}^{n_{60}}\cdots\sum_{n_2=0}^{n_3}\sum_{n_1=0}^{n_2}\sum_{n_0=0}^{n_1}1.\]

Triangle $A_0B_0C_0$ is given in the plane. Consider all triangles $ABC$ such that: (i) The lines $AB,BC,CA$ pass through $C_0,A_0,B_0$, respectvely, (ii) The triangles $ABC$ and $A_0B_0C_0$ are similar. Find the possible positions of the circumcenter of triangle $ABC$.

Let $ABC$ be an acute triangle. Suppose the points $A',B',C'$ lie on its sides $BC,AC,AB$ respectively and the segments $AA',BB',CC'$ intersect in a common point $P$ inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point $P$ and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that $P$ is the orthocenter of the triangle $ABC$. (Grigory Galperin)

A piece of paper is the square $ABCD$. We fold it by placing the vertex $D$ on the point $D' $ of the side $BC$. We assume that $AD$ moves on the segment $A' D'$ and that $A' D' $ intersects $AB$ at $E$. Prove that the perimeter of the triangle $EBD' $ is one half of the perimeter of the square.

Let $N$ and $d$ be two positive integers with $N\geq d+2$. There are $N$ countries connected via two-way direct flights, where each country is connected to exactly $d$ other countries. It is known that for any two different countries, it is possible to go from one to another via several flights. A country is \emph{important} if after removing it and all the $d$ countries it is connected to, there exist two other countries that are no longer connected via several flights. Show that if every country is important, then one can choose two countries so that more than $2d/3$ countries are connected to both of them via direct flights. [i]Proposed by usjl[/i]

Two points $A$ and $B$ lie on two branches of hyperbola given by equation $y=\frac1x$. Let $A_x$ and $A_y$ be projections of $A$ onto coordinate axis, similarly, let $B_x$ and $B_y$ be projections of $B$ onto coordinate axis. Prove that triangles $AB_xB_y$ and $BA_xA_y$ have equal areas.

The point $O{}$ is given in the plane. Find all natural numbers $n{}$ for which $n{}$ points in the plane can be colored red, so that for any two red points $A{}$ and $B{}$ there is a third red point $C{}$ is such that $O{}$ lies strictly inside the triangle $ABC$. [i]From the folklore[/i]

Prove that the series $\sum_p c_p f(px)$, where the summation is over all primes, unconditionally converges in $L^2[0,1]$ for every $1$-periodic function $f$ whose restriction to $[0,1]$ is in $L^2[0,1]$ if and only if $\sum_p |c_p|<\infty$. ([i]Unconditional convergence[/i] means convergence for all rearrangements.) [G. Halasz]

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute the sum $2019 + 201 + 20 + 2$. [b]p2.[/b] The sequence $100, 102, 104,..., 996$ and $998$ is the sequence of all three-digit even numbers. How many three digit even numbers are there? [b]p3.[/b] Find the units digit of $25\times 37\times 113\times 22$. [b]p4.[/b] Samuel has a number in his head. He adds $4$ to the number and then divides the result by $2$. After doing this, he ends up with the same number he had originally. What is his original number? [b]p5.[/b] According to Shay's Magazine, every third president is terrible (so the third, sixth, ninth president and so on were all terrible presidents). If there have been $44$ presidents, how many terrible presidents have there been in total? [b]p6.[/b] In the game Tic-Tac-Toe, a player wins by getting three of his or her pieces in the same row, column, or diagonal of a $3\times 3$ square. How many configurations of $3$ pieces are winning? Rotations and reflections are considered distinct. [b]p7.[/b] Eddie is a sad man. Eddie is cursed to break his arm $4$ times every $20$ years. How many times would he break his arm by the time he reaches age $100$? [b]p8. [/b]The figure below is made from $5$ congruent squares. If the figure has perimeter $24$, what is its area? [img]https://cdn.artofproblemsolving.com/attachments/1/9/6295b26b1b09cacf0c32bf9d3ba3ce76ddb658.png[/img] [b]p9.[/b] Sancho Panza loves eating nachos. If he eats $3$ nachos during the first minute, $4$ nachos during the second, $5$ nachos during the third, how many nachos will he have eaten in total after $15$ minutes? [b]p10.[/b] If the day after the day after the day before Wednesday was two days ago, then what day will it be tomorrow? [b]p11.[/b] Neetin the Rabbit and Poonam the Meerkat are in a race. Poonam can run at $10$ miles per hour, while Neetin can only hop at $2$ miles per hour. If Neetin starts the race $2$ miles ahead of Poonam, how many minutes will it take for Poonam to catch up with him? [b]p12.[/b] Dylan has a closet with t-shirts: $3$ gray, $4$ blue, $2$ orange, $7$ pink, and $2$ black. Dylan picks one shirt at random from his closet. What is the probability that Dylan picks a pink or a gray t-shirt? [b]p13.[/b] Serena's brain is $200\%$ the size of Eric's brain, and Eric's brain is $200\%$ the size of Carlson's. The size of Carlson's brain is what percent the size of Serena's? [b]p14.[/b] Find the sum of the coecients of $(2x + 1)^3$ when it is fully expanded. [b]p15. [/b]Antonio loves to cook. However, his pans are weird. Specifically, the pans are rectangular prisms without a top. What is the surface area of the outside of one of Antonio's pans if their volume is $210$, and their length and width are $6$ and $5$, respectively? [b]p16.[/b] A lattice point is a point on the coordinate plane with $2$ integer coordinates. For example, $(3, 4)$ is a lattice point since $3$ and $4$ are both integers, but $(1.5, 2)$ is not since $1.5$ is not an integer. How many lattice points are on the graph of the equation $x^2 + y^2 = 625$? [b]p17.[/b] Jonny has a beaker containing $60$ liters of $50\%$ saltwater ($50\%$ salt and $50\%$ water). Jonny then spills the beaker and $45$ liters pour out. If Jonny adds $45$ liters of pure water back into the beaker, what percent of the new mixture is salt? [b]p18.[/b] There are exactly 25 prime numbers in the set of positive integers between $1$ and $100$, inclusive. If two not necessarily distinct integers are randomly chosen from the set of positive integers from $1$ to $100$, inclusive, what is the probability that at least one of them is prime? [b]p19.[/b] How many consecutive zeroes are at the end of $12!$ when it is expressed in base $6$? [b]p20.[/b] Consider the following figure. How many triangles with vertices and edges from the following figure contain exactly $1$ black triangle? [img]https://cdn.artofproblemsolving.com/attachments/f/2/a1c400ff7d06b583c1906adf8848370e480895.png[/img] [b]p21.[/b] After Akshay got kicked o the school bus for rowdy behavior, he worked out a way to get home from school with his dad. School ends at $2:18$ pm, but since Akshay walks slowly he doesn't get to the front door until $2:30$. His dad doesn't like to waste time, so he leaves home everyday such that he reaches the high school at exactly $2:30$ pm, instantly picks up Akshay and turns around, then drives home. They usually get home at $3:30$ pm. However, one day Akshay left school early at exactly $2:00$ pm because he was expelled. Trying to delay telling his dad for as long as possible, Akshay starts jogging home. His dad left home at the regular time, saw Akshay on the way, picked him up and turned around instantly. They then drove home while Akshay's dad yelled at him for being a disgrace. They reached home at $3:10$ pm. How long had Akshay been walking before his dad picked him up? [b]p22.[/b] In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Then $\angle BOC = \angle BCD$, $\angle COD =\angle BAD$, $AB = 4$, $DC = 6$, and $BD = 5$. What is the length of $BO$? [b]p23.[/b] A standard six-sided die is rolled. The number that comes up first determines the number of additional times the die will be rolled (so if the first number is $3$, then the die will be rolled $3$ more times). Each time the die is rolled, its value is recorded. What is the expected value of the sum of all the rolls? [b]p24.[/b] Dora has a peculiar calculator that can only perform $2$ operations: either adding $1$ to the current number or squaring the current number. Each minute, Dora randomly chooses an operation to apply to her number. She starts with $0$. What is the expected number of minutes it takes Dora's number to become greater than or equal to $10$? [b]p25.[/b] Let $\vartriangle ABC$ be such that $AB = 2$, $BC = 1$, and $\angle ACB = 90^o$. Let points $D$ and $E$ be such that $\vartriangle ADE$ is equilateral, $D$ is on segment $\overline{BC}$, and $D$ and $E$ are not on the same side of $\overline{AC}$. Segment $\overline{BE}$ intersects the circumcircle of $\vartriangle ADE$ at a second point $F$. If $BE =\sqrt{6}$, find the length of $\overline{BF}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].