Found problems: 85335
We want to find functions $p(t)$, $q(t)$, $f(t)$ such that
(a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$.
(b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$.
(c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$.
Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$.
Find the largest constant $c$, such that if there are $N$ discs in the plane such that every two of them intersect, then there must exist a point which lies in the common intersection of $cN + O(1)$ discs
Let $P$ and $Q$ be non-constant integer-coefficient monic polynomials, and let $a$ and $b$ be integers satisfying $| a | \geq 3$ and $ | b | \geq 3$. These satisfy the following conditions for all positive integers $n$:
$$
P(n) \mid Q(n)^2 + aQ(n) + 1, \quad Q(n) \mid P(n)^2 + bP(n) + 1.
$$
Determine all possible ordered pairs $(a+b, \deg P)$.
[hide=Original wording]
상수다항식이 아닌 최고차항의 계수가 1인 정수계수다항식 $P$, $Q$와 정수 $a$, $b$($| a |, | b | \geq 3$)가 모든 양의 정수 $n$에 대해
$$P(n) \mid Q(n)^2 +aQ(n)+1, \quad Q(n) \mid P(n)^2+bP(n)+1$$
을 만족한다. 이때 가능한 모든 $(a+b, \deg P)$ 순서쌍을 구하여라.
[/hide]
the code system of a new 'MO lock' is a regular $n$-gon,each vertex labelled a number $0$ or $1$ and coloured red or blue.it is known that for any two adjacent vertices,either their numbers or colours coincide.
find the number of all possible codes(in terms of $n$).
Decide if the following statement is true or false:
For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that:
(a) $x_n = a_n + b_n$ for all $n$;
(b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$;
(c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.
2. Let $\Gamma_{1}$ and $\Gamma_{2}$ be externally tangent circles with radii $\frac{1}{2}$ and $\frac{1}{8}$, respectively. The line $\ell$ is a common external tangent to $\Gamma_{1}$ and $\Gamma_{2}$. For $n \geq 3$, we define $\Gamma_{n}$ as the smallest circle tangent to $\Gamma_{n-1}, \Gamma_{n-2}$, and $\ell$. The radius of $\Gamma_{10}$ can be expressed as $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a+b$.
Consider a fixed circle $\Gamma$ with three fixed points $A, B,$ and $C$ on it. Also, let us fix a real number $\lambda \in(0,1)$. For a variable point $P \not\in\{A, B, C\}$ on $\Gamma$, let $M$ be the point on the segment $CP$ such that $CM =\lambda\cdot CP$ . Let $Q$ be the second point of intersection of the circumcircles of the triangles $AMP$ and $BMC$. Prove that as $P$ varies, the point $Q$ lies on a fixed circle.
[i]Proposed by Jack Edward Smith, UK[/i]
Let $x$ be the first term in the sequence $31, 331, 3331, . . .$ which is divisible by $17$. How many digits long is$ x$?
Each of $10$ identical jars contains some milk, up to $10$ percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other $9$ jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most $10$ moves.
[i](4 points)[/i]
On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by $a$ and to the centre of all white squares by $b$. Prove that $a = b$.
(A. Andj ans, Riga)
Let $ABC$ be a triangle with $\angle A=\tfrac{135}{2}^\circ$ and $\overline{BC}=15$. Square $WXYZ$ is drawn inside $ABC$ such that $W$ is on $AB$, $X$ is on $AC$, $Z$ is on $BC$, and triangle $ZBW$ is similar to triangle $ABC$, but $WZ$ is not parallel to $AC$. Over all possible triangles $ABC$, find the maximum area of $WXYZ$.
We are given $m\times n$ table tiled with $3\times 1$ stripes and we are given that $6 | mn$. Prove that there exists a tiling of the table with $2\times 1$ dominoes such that each of these stripes contains one whole domino.
Let $d$ be a positive divisor of $5 + 1998^{1998}$. Prove that $d = 2 \cdot x^2 + 2 \cdot x \cdot y + 3 \cdot y^2$, where $x, y$ are integers if and only if $d$ is congruent to 3 or 7 $\pmod{20}$.
On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.
Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions:
$(1)$Square bar $AB$ is bisected by square bar $CD$
$(2)$At least one square of $AB$ lay on both sides of $CD$
$(3)$At least one square of $CD$ lay on both sides of $AB$
There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet
Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$.
Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.
[b]p1.[/b] A right triangle has hypotenuse of length $12$ cm. The height corresponding to the right angle has length $7$ cm. Is this possible?
[img]https://cdn.artofproblemsolving.com/attachments/0/e/3a0c82dc59097b814a68e1063a8570358222a6.png[/img]
[b]p2.[/b] Prove that from any $5$ integers one can choose $3$ such that their sum is divisible by $3$.
[b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a knight on an arbitrary square. Then the second player can put another knight on a free square that is not controlled by the first knight. Then the first player can put a new knight on a free square that is not controlled by the knights on the board. Then the second player can do the same, etc. A player who cannot put a new knight on the board loses the game. Who has a winning strategy?
[b]p4.[/b] Consider a regular octagon $ABCDEGH$ (i.e., all sides of the octagon are equal and all angles of the octagon are equal). Show that the area of the rectangle $ABEF$ is one half of the area of the octagon.
[img]https://cdn.artofproblemsolving.com/attachments/d/1/674034f0b045c0bcde3d03172b01aae337fba7.png[/img]
[b]p5.[/b] Can you find a positive whole number such that after deleting the first digit and the zeros following it (if they are) the number becomes $24$ times smaller?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
$a_n,b_n,c_n$ are three sequences of positive integers satisfying $$\prod_{d|n}a_d=2^n-1,\prod_{d|n}b_d=\frac{3^n-1}{2},\prod_{d|n}c_d=\gcd(2^n-1,\frac{3^n-1}{2})$$ for all $n\in \mathbb{N}$. Prove that $\gcd(a_n,b_n)|c_n$ for all $n\in \mathbb{N}$
Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be on the lines $AB$ and $AD$, respectively, so that $B$ lies between $A$ and $E$, and $D$ lies between $A$ and $F$. Suppose that $\angle BCE = 20^o$ and $\angle
DCF = 25^o$. Find the area of triangle $\vartriangle EAF$.
Squares $ABKL$, $BCMN$, $CAOP$ are drawn externally on the sides of a triangle $ABC$. The line segments $KL$, $MN$, $OP$, when extended, form a triangle $A'B'C'$. Find the area of $A'B'C'$ if $ABC$ is an equilateral triangle of side length $2$.
On a circular table sit $\displaystyle {n> 2}$ students. First, each student has just one candy. At each step, each student chooses one of the following actions:
(A) Gives a candy to the student sitting on his left or to the student sitting on his right.
(B) Separates all its candies in two, possibly empty, sets and gives one set to the student sitting on his left and the other to the student sitting on his right.
At each step, students perform the actions they have chosen at the same time.
A distribution of candy is called legitimate if it can occur after a finite number of steps.
Find the number of legitimate distributions.
(Two distributions are different if there is a student who has a different number of candy in each of these distributions.)
（Forgive my poor English）
Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.
We call a set of integers $\textit{special}$ if it has $4$ elements and can be partitioned into $2$ disjoint subsets $\{ a,b \}$ and $\{ c, d \}$ such that $ab - cd = 1$. For every positive integer $n$, prove that the set $\{ 1, 2, \dots, 4n \}$ cannot be partitioned into $n$ disjoint special sets.
[i]Proposed by Mohsen Jamali, Iran[/i]
In a triangle $ABC$ with $\widehat{A}=80^{\circ}$ and $\widehat{B}=60^{\circ}$, the internal angle bisector of $\widehat{C}$ meets the side $AB$ at the point $D$. The parallel from $D$ to the side $AC$, meets the side $BC$ at the point $E$.
Find the measure of the angle $\angle EAB$.
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]
[i]Proposed by Thomas Huber, Switzerland[/i]