Do there exist integers $a$ and $b$ such that : (a) the equation $x^2 + ax + b = 0$ has no real roots, and the equation $\lfloor x^2 \rfloor + ax + b = 0$ has at least one real root? [i](2 points)[/i] (b) the equation $x^2 + 2ax + b$ = 0 has no real roots, and the equation $\lfloor x^2 \rfloor + 2ax + b = 0$ has at least one real root? [i]3 points[/i] (By $\lfloor k \rfloor$ we denote the integer part of $k$, that is, the greatest integer not exceeding $k$.) [i]Alexandr Khrabrov[/i]

Let $f:[1,\infty)\to(0,\infty)$ be a continuous function. Assume that for every $a>0$, the equation $f(x)=ax$ has at least one solution in the interval $[1,\infty)$. (a) Prove that for every $a>0$, the equation $f(x)=ax$ has infinitely many solutions. (b) Give an example of a strictly increasing continuous function $f$ with these properties.

A [i]smooth number[/i] is a positive integer of the form $2^m 3^n$, where $m$ and $n$ are nonnegative integers. Let $S$ be the set of all triples $(a, b, c)$ where $a$, $b$, and $c$ are smooth numbers such that $\gcd(a, b)$, $\gcd(b, c)$, and $\gcd(c, a)$ are all distinct. Evaluate the infinite sum $\sum_{(a,b,c) \in S} \frac{1}{abc}$. Recall that $\gcd(x, y)$ is the greatest common divisor of $x$ and $y$.

On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial on the blackboard has $37$ distinct positive roots. [i](8 points)[/i] [i]Alexandr Kuznetsov[/i]

A school needs to elect its president. The school has $121$ students, each of whom belongs to one of two tribes: Geometers or Algebraists. Two candidates are running for president: one Geometer and one Algebraist. The Geometers vote only for Geometers and the Algebraists only for Algebraists. There are more Algebraists than Geometers, but the Geometers are resourceful. They convince the school that the following two-step procedure is fairer: [list=a] [*]The school is divided into $11$ groups, with $11$ students in each group. Each group elects a representative for step 2. [*]The $11$ elected representatives elect a president. [/list] Not only do the Geometers manage to have this two-step procedure approved, they also volunteer to assign the students to groups for step 1. What is the minimum number of Geometers in the school that guarantees they can elect a Geometer as president? (In any stage of voting, the majority wins.)

Prove the inequality $$ 1^k+2^k+...+n^k \le \frac{n^{2k}-(n-1)^k}{n^k-(n-1)^k}$$ (L Emelianov)

Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome—it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period? $\textbf{(A) }50 \qquad \textbf{(B) }55 \qquad \textbf{(C) }60\qquad \textbf{(D) }65\qquad\textbf{(E) }70$

In a $ 4 \times 4 $ grid of sixteen unit squares, exactly $8$ are shaded so that each shaded square shares an edge with exactly one other shaded square. How many ways can this be done?

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

Let $ABC$ be a triangle with centroid $T$. Denote by $M$ the midpoint of $BC$. Let $D$ be a point on the ray opposite to the ray $BA$ such that $AB = BD$. Similarly, let $E$ be a point on the ray opposite to the ray $CA$ such that $AC = CE$. The segments $T D$ and $T E$ intersect the side $BC$ in $P$ and $Q$, respectively. Show that the points $P, Q$ and $M$ split the segment $BC$ into four parts of equal length.

Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $ 100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks $ 400$ feet or less to the new gate be a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

What is the smallest perfect square that ends in $9009$?

Let $\mathbf{P}_1,\mathbf{P}_2$ be convex polygons with perimeters $o_1,o_2,$ respectively. Show that if $\mathbf P_1\subseteq\mathbf P_2,$ then $o_1\le o_2.$

Consider the numbers arranged in the following way: \[\begin{array}{ccccccc} 1 & 3 & 6 & 10 & 15 & 21 & \cdots \\ 2 & 5 & 9 & 14 & 20 & \cdots & \cdots \\ 4 & 8 & 13 & 19 & \cdots & \cdots & \cdots \\ 7 & 12 & 18 & \cdots & \cdots & \cdots & \cdots \\ 11 & 17 & \cdots & \cdots & \cdots & \cdots & \cdots \\ 16 & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\] Find the row number and the column number in which the the number $20096$ occurs.

How many four-digit positive integers have at least one digit that is a 2 or a 3? $ \textbf{(A) } 2439 \qquad \textbf{(B) } 4096 \qquad \textbf{(C) } 4903 \qquad \textbf{(D) } 4904 \qquad \textbf{(E) } 5416$

Let $ABC$ be an acute-angled triangle such that $AB+BC=4AC$. Let $D$ in $AC$ such that $BD$ is angle bisector of $\angle ABC$. In the segment $BD$, points $P$ and $Q$ are marked such that $BP=2DQ$. The perpendicular line to $BD$, passing by $Q$, cuts the segments $AB$ and $BC$ in $X$ and $Y$, respectively. Let $L$ be the parallel line to $AC$ passing by $P$. The point $B$ is in a different half-plane(with respect to the line $L$) of the points $X$ and $Y$. An ant starts a run in the point $X$, goes to a point in the line $AC$, after that goes to a point in the line $L$, returns to a point in the line $AC$ and finishes in the point $Y$. Prove that the least length of the ant's run is equal to $4XY$.

Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.

Determine the radius of a sphere $S$ that passes through the centroids of each face of a given tetrahedron $T$ inscribed in a unit sphere with center $O$. Also, determine the distance from $O$ to the center of $S$ as a function of the edges of $T.$

For positive integers $a$ and $b$, let $M(a,b) = \tfrac{\text{lcm}(a,b)}{\gcd(a,b)},$ and for each positive integer $n \ge 2,$ define \[x_n = M(1, M(2, M(3, \dots , M(n - 2, M(n - 1, n))\cdots))).\] Compute the number of positive integers $n$ such that $2 \le n \le 2021$ and $5x_n^2 + 5x_{n+1}^2 = 26x_nx_{n+1}.$

Let $x$ and $y$ be two positive real numbers, such that $x + y = 1$. Prove that $$\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9$$

Determine $ m > 0$ so that $ x^4 \minus{} (3m\plus{}2)x^2 \plus{} m^2 \equal{} 0$ has four real solutions forming an arithmetic series: i.e., that the solutions may be written $ a, a\plus{}b, a\plus{}2b,$ and $ a\plus{}3b$ for suitable $ a$ and $ b$. A. 1 B. 3 C. 7 D. 12 E. None of these

We call a positive integer [i]good number[/i], if it is divisible by squares of all its prime factors. Show that there are infinitely many pairs of consequtive numbers both are [i]good[/i].

(i) Find a formula for $S_n = -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^n n^2 \times (n + 1)$ in terms of the positive integer $n$. Justify your answer. (As an example, one has $1 + 2 + 3 +...+n = \frac{n(n+1)}{2}$) (ii) Using your formula in (i), find the value of $ -1^2 \times 2 + 2^2 \times 3 - 3^2 \times 4 + 4^2 \times 5 -... + (-l)^{100} 100^2 \times (100 + 1)$

Find the number of positive integers x satisfying the following two conditions: 1. $x<10^{2006}$ 2. $x^{2}-x$ is divisible by $10^{2006}$

Find all positive integers $ n$ for which there exist integers $ a,b,c$ such that $ a\plus{}b\plus{}c\equal{}0$ and the number $ a^{n}\plus{}b^{n}\plus{}c^{n}$ is prime.