Found problems: 85335
Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$. Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$.
(translated by L. Erdős)
Which of the following statements are true?
(A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect.
(B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b.
(C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.
A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$. Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area.
Let $A$ and $B$ the series: $$A=\sum \limits_{n=1}^{\infty}\frac{C_{2n}^1}{C_{2n}^0+C_{2n}^1+\cdots +C_{2n}^{2n}},\ B=\sum \limits_{n=1}^{\infty}\frac{\Gamma \left(n+\frac{1}{2}\right) }{\Gamma \left(n+\frac{5}{2}\right)}$$Study if $\frac{A}{B}$ is irrational number.
The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that
\[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]
Let $A, B,C$ be colinear points in this order, $\omega$ an arbitrary circle passing through $B$ and $C$, and $l$ an arbitrary line different from $BC$, passing through A and intersecting $\omega$ at $M$ and $N$. The bisectors of the angles $\angle CMB$ and $\angle CNB$ intersect $BC$ at $P$ and $Q$. Prove that $AP\cdot AQ = AB \cdot AC$.
For each positive integer $n$ we consider the sequence of $2004$ integers$$\left [n+\sqrt{n}\right ],\left [n+1+\sqrt{n+1}\right ],\left [n+2+\sqrt{n+2}\right ],\ldots ,\left [n+2003+\sqrt{n+2003}\right ]$$Determine the smallest integer $n$ such that the $2004$ numbers in the sequence are $2004$ consecutive integers.
Clarification: The brackets indicate the integer part.
In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$
The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$
A parabola and a hyperbola are drawn on the coordinate plane. The graphs intersect at three points $A, B, C$ and the axis of the parabola is the asymptote of the hyperbola. Prove that the intersection point of the medians of the triangle $ABC$ lies on the axis of the parabola.
[i]From the folklore[/i]
In $\vartriangle ABC, D$ and $E$ are two points on segment $BC$ such that $BD = CE$ and $\angle BAD = \angle CAE$. Prove that $\vartriangle ABC$ is isosceles
Ivan has a $n \times n$ board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let $d_n$ be the maximum number of black squares possible, prove that there exist some real constants $a$, $b$, $c\ge 0$ such that; $$an^2-bn\le d_n\le an^2+cn.$$
[i]Proposed by Ivan Chan Kai Chin[/i]
Let $ S $ be a nonempty subset of a finite group $ G, $ and $ \left( S^j \right)_{j\ge 1} $ be a sequence of sets defined as $ S^j=\left.\left\{\underbrace{xy\cdots z}_{\text{j terms}} \right| \underbrace{x,y,\cdots ,z}_{\text{j terms}} \in S \right\} . $ Prove that:
[b]a)[/b] $ \exists i_0\in\mathbb{N}^*\quad i\ge i_0\implies \left| S^i\right| =\left| S^{1+i}\right| $
[b]b)[/b] $ S^{|G|}\le G $
Solve in real numbers the equation
\[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]
Let $ABCD$ be a convex quadrilateral. $AC$ and $BD$ meet at $P$, with $\angle APD=60^{\circ}$. Let $E,F,G$, and $H$ be the midpoints of $AB,BC,CD$ and $DA$ respectively. Find the greatest positive real number $k$ for which
\[EG+3HF\ge kd+(1-k)s \]
where $s$ is the semi-perimeter of the quadrilateral $ABCD$ and $d$ is the sum of the lengths of its diagonals. When does the equality hold?
Three numbers are selected at random from the interval $[0,1]$. What is the probability that they form the lengths of the sides of a triangle?
[b]p25.[/b] You are given that $1000!$ has $2568$ decimal digits. Call a permutation $\pi$ of length $1000$ good if $\pi(2i) > \pi (2i - 1)$ for all $1 \le i \le 500$ and $\pi (2i) > \pi (2i + 1)$ for all $1 \le i \le 499$. Let $N$ be the number of good permutations. Estimate $D$, the number of decimal digits in $N$.
You will get $\max \left( 0, 25 - \left\lceil \frac{|D-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer.
[b]p26.[/b] A year is said to be [i]interesting [/i] if it is the product of $3$, not necessarily distinct, primes (for example $2^2 \cdot 5$ is interesting, but $2^2 \cdot 3 \cdot 5$ is not). How many interesting years are there between $ 5000$ and $10000$, inclusive?
For an estimate of $E$, you will get $\max \left( 0, 25 - \left\lceil \frac{|E-X|}{10} \right\rceil \right)$ points, where $X$ is the true answer.
[b]p27.[/b] Sam chooses $1000$ random lattice points $(x, y)$ with $1 \le x, y \le 1000$ such that all pairs $(x, y)$ are distinct. Let $N$ be the expected size of the maximum collinear set among them. Estimate $\lfloor 100N \rfloor$. Let $S$ be the answer you provide and $X$ be the true value of $\lfloor 100N \rfloor$. You will get $\max \left( 0, 25 - \left\lceil \frac{|S-X|}{10} \right\rceil \right)$ points for your estimate.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A pair of positive integers $(a,b)$ is called [i]charrua[/i] if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called [i]non-charrua[/i].
a) Prove that there are infinite [i]non-charrua[/i] pairs.
b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is [i]charrua[/i].
How many integers less than $400$ have exactly $3$ factors that are perfect squares?
Jorge's teacher asks him to plot all the ordered pairs $ (w, l)$ of positive integers for which $ w$ is the width and $ l$ is the length of a rectangle with area 12. What should his graph look like?
$ \textbf{(A)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,12));
dot((2,6));
dot((3,4));
dot((4,3));
dot((6,2));
dot((12,1));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(B)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,1));
dot((3,3));
dot((5,5));
dot((7,7));
dot((9,9));
dot((11,11));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(C)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,11));
dot((3,9));
dot((5,7));
dot((7,5));
dot((9,3));
dot((11,1));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(D)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((1,6));
dot((3,6));
dot((5,6));
dot((7,6));
dot((9,6));
dot((11,6));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
$ \textbf{(E)}$[asy]size(75);
draw((0,-1)--(0,13));
draw((-1,0)--(13,0));
dot((6,1));
dot((6,3));
dot((6,5));
dot((6,7));
dot((6,9));
dot((6,11));
label("$l$", (0,6), W);
label("$w$", (6,0), S);[/asy]
Is it possible to find a set of $100$ (or $200$) points on the boundary of a cube such that this set remains fixed under all rotations which leave the cube fixed ?
Determine the number of functions $f$ from the integers to $\{1,2,\cdots,15\}$ which satisfy $$f(x)=f(x+15)$$
and
$$f(x+f(y))=f(x-f(y))$$
for all $x,y$.
[i]Proposed by Vijay Srinivasan[/i]
Answer either (i) or (ii):
(i) Prove that
$$\sum_{n=2}^{\infty} \frac{\cos (\log \log n)}{\log n}$$
diverges.
(ii) Assume that $p>0, a>0$, and $ac-b^{2} >0,$ and show that
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{ dx\; dy}{(p+ax^2 +2bxy+ cy^2 )^{2}}= \pi p^{-1} (ac-b^{2})^{- 1\slash 2}.$$