Found problems: 85335
$ A$ and $ B$ play the following game with a polynomial of degree at least 4:
\[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0
\]
$ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
What is the smallest integer $n$ for which any subset of $\{1,2,3,\ldots,20\}$ of size $n$ must contain two numbers that differ by $8$?
$\textbf{(A) }2\qquad\textbf{(B) }8\qquad\textbf{(C) }12\qquad\textbf{(D) }13\qquad\textbf{(E) }15$
Show that if $ \left| ax^2+bx+c\right|\le 1, $ for all $ x\in [-1,1], $ then $ |a|+|b|+|c|\le 4. $
Let $ \left( a_n\right)_{n\ge 1} $ be a sequence defined by $ a_1>0 $ and $ \frac{a_{n+1}}{a}=\frac{a_n}{a}+\frac{a}{a_n} , $ with $ a>0. $ Calculate $ \lim_{n\to\infty} \frac{a_n}{\sqrt{n+a}} . $
[i]Florin Rotaru[/i]
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)
Pentagon $ABCDE$ has $AB = BC = CD = DE$, $\angle ABC = \angle BCD = 108^o$, and $\angle CDE = 168^o$. Find the measure of angle $\angle BEA$ in degrees.
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection.
Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $ABCD$ be a cyclic quadrilateral and $M,N$ be the midpoints of $AB$, $CD$ respectively. The diagonals $AC$ and $BD$ intersect at $L$. Suppose that the circumcircle of $LMN$, with center $T$, intersects the circumcircle of $ABCD$ at two distinct points $X,Y$. If the line $MN$ intersects the line $XY$ at $S$ and the line $XM$ intersects the line $YN$ at $P$, prove that $PL$ is perpendicular to $ST$.
Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example, to form multidigit numbers (such as $222$) or powers (such as $2^2$).
Valid examples: $$\left((2\times 2+2)\times 2-\frac{2}{2}\right)\times 2=22 \;\;, \;\; (2\times2\times 2-2)\times \left(2\times 2 +\frac{2+2+2}{2}\right)=42$$
[i]Proposed by Stephan Wagner, Austria[/i]
Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3).$ If Sophie is at $(x,y),$ in a single step she can move to one of $(x +1,y), (x,y + 1), (x - 1,y +1),$ or $(x +1,y -1).$ She cannot revisit any points along her path, and neither her $x$-coordinate nor her $y$-coordinate can ever be less than $0$ or greater than $3.$ Compute the number of ways for Sophie to reach $(3,3).$
It is necessary to construct an angle whose sine is three times greater than its cosine. Describe how this can be done.
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
Last year, the U.S. House of Representatives passed a bill which would make Washington, D.C. into the $51$st state. Naturally, the mathematicians are upset that Congress won’t prioritize mathematical interest of flag design in choosing how many U.S. states there should be. Suppose the U.S. flag must contain, as it does now, stars arranged in rows alternating between $n$ and $n - 1$ stars, starting and ending with rows of n stars, where $n \ge 2$ is some integer and the flag has more than one row. What is the minimum number of states that the U.S. would need to contain so that there are at least three different ways, excluding rotations, to arrange the stars on the flag?
Find all non-negative integers $x, y$ and $z$ satisfying the equation: \[7^{x}+1=3^{y}+5^z\]
Let $A, B \in M_6 (\mathbb{Z} )$ such that $A \equiv I \equiv B \text{ mod }3$ and $A^3 B^3 A^3 = B^3$. Prove that $A = I$. Here $M_6 (\mathbb{Z} )$ indicates the $6$ by $6$ matrices with integer entries, $I$ is the identity matrix, and $X \equiv Y \text{ mod }3$ means all entries of $X-Y$ are divisible by $3$.
Let $a>k$ be natural numbers and $r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n$ be sequences of natural numbers such that:
$(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)$
Prove that these sequences are equal.
Let $a,\ b$ be real numbers. If $\int_0^3 (ax-b)^2dx\leq 3$ holds, then find the values of $a,\ b$ such that $\int_0^3 (x-3)(ax-b)dx$ is minimized.
Let $K$ and $L$ be positive integers. On a board consisting of $2K \times 2L$ unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbouring square. It never visits a square twice. At the end some squares may remain unvisited.
In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle [i]MEMOrable[/i].
Determine the number of different MEMOrable rectangles.
[i]Remark: Rectangles are different unless they consist of exactly the same squares.[/i]
Let $ABCD$ be a convex quadrilateral and let $M$, $N$, $P$ and $Q$ be the midpoints of the sides $AB$, $CD$, $BC$ and $DA$ respectively. The line $MN$ intersects the segments $AP$ and $CQ$ at points $X$ and $Y$, respectively. Suppose that $MX = NY$. Prove that $\text{area}(ABCD) = 4 \cdot \text{area}(BXDY).$
Let $k$ be a real number. Define on the set of reals the operation $x*y$= $\frac{xy}{x+y+k}$ whenever $x+y$ does not equal $-k$. Let $x_1<x_2<x_3<x_4$ be the roots of $t^4=27(t^2+t+1)$.suppose that $[(x_1*x_2)*x_3]*x_4=1$. Find all possible values of $k$
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]