Sonia has $10$, $15$ and $20$ cent stamps with total face value of $\$5$. She has $30$ stamps altogether. Prove that she has more $20$ cent stamps than $10$ cent stamps.

Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.

Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)

Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The infinite series $$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$ has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $a, b, c, d \in [0, 1]$. Prove that $$\frac{a}{1 + b}+\frac{b}{1 + c}+\frac{c}{1 + d}+\frac{d}{1 + a}+ abcd \le 3.$$

Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$ $$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$ $$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$ [i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$, with $n\geq 2$. Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \] [i]Romania[/i]

Solve \[\left\{ \begin{array}{l} y(x+y)^2 = 9 \\ y(x^3-y^3) = 7 \\ \end{array} \right. \]

The set of polynomials $f_1, f_2, \ldots, f_n$ with real coefficients is called [i]special [/i], if for any different $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial $\dfrac{2}{3}f_i + f_j + f_k$ has no real roots, but for any different $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root. a) Give an example of a [i]special [/i] set of four polynomials whose sum is not a zero polynomial. b) Is there a [i]special [/i] set of five polynomials?

Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.

Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$

Find the sum of all integers from $1$ to $1000$ inclusive which contain at least one $7$ in their digits, i.e. find $$7 + 17 +... + 979 + 987 + 997.$$

[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] [u]Set 1[/u] [b]D.1 / L.1[/b] Find the units digit of $3^{1^{3^{3^7}}}$. [b]D.2[/b] Find the positive solution to the equation $x^3 - x^2 = x - 1$. [b]D.3[/b] Points $A$ and $B$ lie on a unit circle centered at O and are distance $1$ apart. What is the degree measure of $\angle AOB$? [b]D.4[/b] A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between $1$ and $2019$, inclusive? [b]D.5[/b] Ted has four children of ages $10$, $12$, $15$, and $17$. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now? [u]Set 2[/u] [b]D.6[/b] Mr. Schwartz is on the show Wipeout, and is standing on the first of $5$ balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is $1/2$, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish. [b]D.7 / L. 5[/b] Kevin has written $5$ MBMT questions. The shortest question is $5$ words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question? [b]D.8 / L. 3[/b] Square $ABCD$ with side length $1$ is rolled into a cylinder by attaching side $AD$ to side $BC$. What is the volume of that cylinder? [b]D.9 / L.4[/b] Haydn is selling pies to Grace. He has $4$ pumpkin pies, $3$ apple pies, and $1$ blueberry pie. If Grace wants $3$ pies, how many different pie orders can she have? [b]D.10[/b] Daniel has enough dough to make $8$ $12$-inch pizzas and $12$ $8$-inch pizzas. However, he only wants to make $10$-inch pizzas. At most how many $10$-inch pizzas can he make? [u]Set 3[/u] [b]D.11 / L.2[/b] A standard deck of cards contains $13$ cards of each suit (clubs, diamonds, hearts, and spades). After drawing $51$ cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs? [b]D.12 / L. 7[/b] Let $s(n)$ be the sum of the digits of $n$. Let $g(n)$ be the number of times s must be applied to n until it has only $1$ digit. Find the smallest n greater than $2019$ such that $g(n) \ne g(n + 1)$. [b]D.13 / L. 8[/b] In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the $2019$ contestants, is secretly told that her score is $S$. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of $S$? [b]D.14 / L. 9[/b] Let $A$ and $B$ be opposite vertices on a cube with side length $1$, and let $X$ be a point on that cube. Given that the distance along the surface of the cube from $A$ to $X$ is $1$, find the maximum possible distance along the surface of the cube from $B$ to $X$. [b]D.15[/b] A function $f$ with $f(2) > 0$ satisfies the identity $f(ab) = f(a) + f(b)$ for all $a, b > 0$. Compute $\frac{f(2^{2019})}{f(23)}$. PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$ We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$. Prove that for every $n>0, y_n$ is the square of an odd integer.

Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations $x^5 = 2y^3 + y - 2$ $y^5 = 2z^3 + z - 2$ $z^5 = 2x^3 + x - 2$

Let $a$, $b$, $c$ and $d$ be positive real numbers with $a+b+c+d = 4$. Prove that $\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2$.

Let $a=x_1\leq x_2\leq ...\leq x_n=b$. Prove the following inequality: $(x_1+x_2+...+x_n )(\frac{1}{x_1} +\frac{1}{x_2} +...+\frac{1}{x_n} )\leq \frac{(a+b)}{4ab} n^2$.

A sequence of non-negative rational numbers $ a(1), a(2), a(3), \ldots$ satisfies $ a(m) \plus{} a(n) \equal{} a(mn)$ for arbitrary natural $ m$ and $ n$. Show that not all elements of the sequence can be distinct.

Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively, other than the vertices. Prove that $\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$

[b]p1.[/b] To convert between Fahrenheit, $F$, and Celsius, $C$, the formula is $F = \frac95 C + 32$. Jennifer, having no time to be this precise, instead approximates the temperature of Fahrenheit, $\widehat F$, as $\widehat F = 2C + 30$. There is a range of temperatures $C_1 \le C \le C_2$ such that for any $C$ in this range, $| \widehat F - F| \le 5$. Compute the ordered pair $(C_1,C_2)$. [b]p2.[/b] Compute integer $x$ such that $x^{23} = 27368747340080916343$. [b]p3.[/b] The number of ways to flip $n$ fair coins such that there are no three heads in a row can be expressed with the recurrence relation $$ S(n + 1) = a_0 S(n) + a_1 S(n - 1) + ... + a_k S(n - k) $$ for sufficiently large $n$ and $k$ where $S(n)$ is the number of valid sequences of length $n$. What is $\sum^k_{n=0}|a_n|$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds: \[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\] Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$ \[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]