Find all triples $a,b,c$ of positive integers such that \[ ( 1 + \frac{1}{a} ) ( 1 + \frac{1}{b}) ( 1 + \frac{1}{c} ) = 3. \]

Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$. (15 points)

Determine all real solutions of the following system of equations: $$x+y^2=y^3$$ $$y+x^2=x^3$$

Let $n$ and $k$ be natural numbers and $a_1,a_2,\ldots ,a_n$ be positive real numbers satisfying $a_1+a_2+\cdots +a_n=1$. Prove that \[\dfrac {1} {a_1^{k}}+\dfrac {1} {a_2^{k}}+\cdots +\dfrac {1} {a_n^{k}} \ge n^{k+1}.\]

Determine the sixth number after the decimal point in the number $(\sqrt{1978} +\lfloor\sqrt{1978}\rfloor)^{20}$

Niek has $16$ square cards that are yellow on one side and red on the other. He puts down the cards to form a $4 \times 4$-square. Some of the cards show their yellow side and some show their red side. For a colour pattern he calculates the [i]monochromaticity [/i] as follows. For every pair of adjacent cards that share a side he counts $+1$ or $-1$ according to the following rule: $+1$ if the adjacent cards show the same colour, and $-1$ if the adjacent cards show different colours. Adding this all together gives the monochromaticity (which might be negative). For example, if he lays down the cards as below, there are $15$ pairs of adjacent cards showing the same colour, and $9$ such pairs showing different colours. [asy] unitsize(1 cm); int i; fill(shift((0,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((0,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((2,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,1))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((1,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((0,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((2,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), yellow); fill(shift((3,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); for (i = 0; i <= 4; ++i) { draw((i,0)--(i,4)); draw((0,i)--(4,i)); } [/asy] The monochromaticity of this pattern is thus $15 \cdot (+1) + 9 \cdot (-1) = 6$. Niek investigates all possible colour patterns and makes a list of all possible numbers that appear at least once as a value of the monochromaticity. That is, Niek makes a list with all numbers such that there exists a colour pattern that has this number as its monochromaticity. (a) What are the three largest numbers on his list? ([i]Explain your answer. If your answer is, for example, $ 12$, $9$ and $6$, then you have to show that these numbers do in fact appear on the list by giving a colouring for each of these numbers, and furthermore prove that the numbers $7$, $ 8$, $10$, $11$ and all numbers bigger than $ 12$ do not appear.[/i]) (b) What are the three smallest (most negative) numbers on his list? (c) What is the smallest positive number (so, greater than $0$) on his list?

Determine the leftmost three digits of the number \[1^1+2^2+3^3+...+999^{999}+1000^{1000}.\]

What least possible number of unit squares $(1\times1)$ must be drawn in order to get a picture of $25 \times 25$-square divided into $625$ of unit squares?

Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]

Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where \[ A = \prod_{(a,b) \in S} a , \quad B = \prod_{(a,b) \in S} b , \quad \text{and} \quad C = \sum_{(a,b) \in S} ab. \][i]Proposed by Evan Chen[/i]

Suppose that a function $f(x)$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii). (i) $f'(x)$ is continuous. (ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$. (iii) $f(0)=-1$ Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ holds for $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, then find $f(x)$. [i]1975 Waseda University entrance exam/Science and Technology[/i]

Georg has $2n + 1$ cards with one number written on each card. On one card the integer $0$ is written, and among the rest of the cards, the integers $k = 1, ... , n$ appear, each twice. Georg wants to place the cards in a row in such a way that the $0$-card is in the middle, and for each $k = 1, ... , n$, the two cards with the number $k$ have the distance $k$ (meaning that there are exactly $k - 1$ cards between them). For which $1 \le n \le 10$ is this possible?

Find the largest positive integer $n$ of the form $n=p^{2\alpha}q^{2\beta}r^{2\gamma}$ for primes $p<q, r$ and positive integers $\alpha, \beta, \gamma$, such that $|r-pq|=1$ and $p^{2\alpha}-1, q^{2\beta}-1, r^{2\gamma}-1$ all divide $n$.

Given a regular hexagon, a circle is drawn circumscribing it and another circle is drawn inscribing it. The ratio of the area of the larger circle to the area of the smaller circle can be written in the form $\frac{m}{n}$ , where m and n are relatively prime positive integers. Compute $m + n$.

Compute the value of $$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$ Your answer should not involve any trigonometric functions. [i]Proposed by Howard Halim[/i]

Find all positive integers $n$ for which $$S(n^2)+S(n)^2=n$$ where $S(m)$ denotes the sum of digits of $m$.

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.

A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles. [i]A. Shapovalov[/i]

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions: (i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers); (ii) $|a_1-b_1|+|a_2-b_2|=2010$; (iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$; (iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$. [i]Massimo Gobbino, Italy[/i]

For positive integers $n, k, r$, denote by $A(n, k, r)$ the number of integer tuples $(x_1, x_2, \ldots, x_k)$ satisfying the following conditions. [list] [*] $x_1 \ge x_2 \ge \cdots \ge x_k \ge 0$ [*] $x_1+x_2+ \cdots +x_k = n$ [*] $x_1-x_k \le r$ [/list] For all positive integers $s, t \ge 2$, prove that $$A(st, s, t) = A(s(t-1), s, t) = A((s-1)t, s, t).$$

Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\pm 1$ from some digit further to the left. (Your answer should be an explicit function of $n$ in simplest form.)

[b]p1.[/b] How many two-digit primes have a units digit of $3$? [b]p2.[/b] How many ways can you arrange the letters $A$, $R$, and $T$ such that it makes a three letter combination? Each letter is used once. [b]p3.[/b] Hanna and Kevin are running a $100$ meter race. If Hanna takes $20$ seconds to finish the race and Kevin runs $15$ meters per second faster than Hanna, by how many seconds does Kevin finish before Hanna? [b]p4.[/b] It takes an ant $3$ minutes to travel a $120^o$ arc of a circle with radius $2$. How long (in minutes) would it take the ant to travel the entirety of a circle with radius $2022$? [b]p5.[/b] Let $\vartriangle ABC$ be a triangle with angle bisector $AD$. Given $AB = 4$, $AD = 2\sqrt2$, $AC = 4$, find the area of $\vartriangle ABC$. [b]p6.[/b] What is the coefficient of $x^5y^2$ in the expansion of $(x + 2y + 4)^8$? [b]p7.[/b] Find the least positive integer $x$ such that $\sqrt{20475x}$ is an integer. [b]p8.[/b] What is the value of $k^2$ if $\frac{x^5 + 3x^4 + 10x^2 + 8x + k}{x^3 + 2x + 4}$ has a remainder of $2$? [b]p9.[/b] Let $ABCD$ be a square with side length $4$. Let $M$, $N$, and $P$ be the midpoints of $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$, respectively. The area of the intersection between $\vartriangle DMN$ and $\vartriangle ANP$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p10.[/b] Let $x$ be all the powers of two from $2^1$ to $2^{2023}$ concatenated, or attached, end to end ($x = 2481632...$). Let y be the product of all the powers of two from $2^1$ to $2^{2023}$ ($y = 2 \cdot 4 \cdot 8 \cdot 16 \cdot 32... $ ). Let 2a be the largest power of two that divides $x$ and $2^b$ be the largest power of two that divides $y$. Compute $\frac{b}{a}$ . [b]p11.[/b] Larry is making a s’more. He has to have one graham cracker on the top and one on the bottom, with eight layers in between. Each layer can made out of chocolate, more graham crackers, or marshmallows. If graham crackers cannot be placed next to each other, how many ways can he make this s’more? [b]p12.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, $AC = 5$. Circle $O$ is centered at $B$ and has radius $\frac{8\sqrt{3}}{5}$ . The area inside the triangle but not inside the circle can be written as $\frac{a-b\sqrt{c}-d\pi}{e}$ , where $gcd(a, b, d, e) =1$ and $c$ is squarefree. Find $a + b + c + d + e$. [b]p13.[/b] Let $F(x)$ be a quadratic polynomial. Given that $F(x^2 - x) = F (2F(x) - 1)$ for all $x$, the sum of all possible values of $F(2022)$ can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p14.[/b] Find the sum of all positive integers $n$ such that $6\phi (n) = \phi (5n)+8$, where $\phi$ is Euler’s totient function. Note: Euler’s totient $(\phi)$ is a function where $\phi (n)$ is the number of positive integers less than and relatively prime to $n$. For example, $\phi (4) = 2$ since only $1$, $3$ are the numbers less than and relatively prime to $4$. [b]p15.[/b] Three numbers $x$, $y$, and $z$ are chosen at random from the interval $[0, 1]$. The probability that there exists an obtuse triangle with side lengths $x$, $y$, and $z$ can be written in the form $\frac{a\pi-b}{c}$ , where $a$, $b$, $c$ are positive integers with $gcd(a, b, c) = 1$. Find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number whose reciprocal is the sum of the reciprocal of $9 + 15i$ and the reciprocal of $9-15i$ .

For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.