Found problems: 288
Given two distinct elements $a,b \in \{-1,0,1\}$, consider the matrix $A$ .
Find a subset $S$ of the set of the rows of $A$, of minimum size, such that every other row of $A$ is a linear combination of the rows in $S$ with integer coefficients.
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$
If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?
For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?
In a right-angled triangle in which all side lengths are integers, one has a cathetus length $1994$. Determine the length of the hypotenuse.
Suppose that $a$ and $b$ are integers. Prove that there are integers $c$ and $d$ such that $a+b+c+d=0$ and $ac+bd=0$, if and only if $a-b$ divides $2ab$.
Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.
Find the smallest real number $A$, such that there are two different triangles, with integer sidelengths and so that the area of each be $A$.
For which positive integers $n$ is the following statement true:
if $a_1, a_2, ... , a_n$ are positive integers, $a_k \le n$ for all $k$ and $\sum\limits_{k=1}^{{n}}{a_k}=2n$
then it is always possible to choose $a_{i1} , a_{i2} , ..., a_{ij}$ in such a way that
the indices $i_1, i_2,... , i_j$ are different numbers, and $\sum\limits_{k=1}^{{{j}}}{a_{ik}}=n$?
Consider a triangle $XYZ$ and a point $O$ in its interior. Three lines through $O$ are drawn, parallel to the respective sides of the triangle. The intersections with the sides of the triangle determine six line segments from $O$ to the sides of the triangle. The lengths of these segments are integer numbers $a, b, c, d, e$ and $f$ (see figure).
Prove that the product $a \cdot b \cdot c\cdot d \cdot e \cdot f$ is a perfect square.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, O, X, Y, Z;
X = (1,4);
Y = (0,0);
Z = (5,1.5);
O = (1.8,2.2);
A = extension(O, O + Z - X, X, Y);
B = extension(O, O + Y - Z, X, Y);
C = extension(O, O + X - Y, Y, Z);
D = extension(O, O + Z - X, Y, Z);
E = extension(O, O + Y - Z, Z, X);
F = extension(O, O + X - Y, Z, X);
draw(X--Y--Z--cycle);
draw(A--D);
draw(B--E);
draw(C--F);
dot("$A$", A, NW);
dot("$B$", B, NW);
dot("$C$", C, SE);
dot("$D$", D, SE);
dot("$E$", E, NE);
dot("$F$", F, NE);
dot("$O$", O, S);
dot("$X$", X, N);
dot("$Y$", Y, SW);
dot("$Z$", Z, dir(0));
label("$a$", (A + O)/2, SW);
label("$b$", (B + O)/2, SE);
label("$c$", (C + O)/2, SE);
label("$d$", (D + O)/2, SW);
label("$e$", (E + O)/2, SE);
label("$f$", (F + O)/2, NW);
[/asy]
Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.
Product of two integers is $1$ less than three times of their sum. Find those integers.
Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.
Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.
Consider the set of all pairs of positive integers $(A , B)$ in which $A < B$ . Some of these pairs are to $be$ designated as "black" , while the remainder are to be designated as "white" . Is it possible to designate these pairs in such a way that for any triple of positive integers of form $A, A + D, A + 2D$, in which $D > 0$, the associated pairs $(A, A + D )$ , $(A , A + 2D)$ and $(A + D, A + 2D)$ would include at least one pair of each colour?
A sequence of integers $a_1,a_2,\ldots $ is defined by $a_1=1,a_2=2$ and for $n\geq 1$,
$$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. $$
(a) Prove that the sequence contains infinitely many positive terms and infinitely many negative terms.
(b) Prove that no term of the sequence is zero.
(c) Show that if $n = 2^k - 1$ for $k\geq 2$, then $a_n$ is divisible by $7$.
Let $a, b \ge 2$ be positive integers with $gcd (a, b) = 1$. Let $r$ be the smallest positive value that $\frac{a}{b}- \frac{c}{d}$ can take, where $c$ and $d$ are positive integers satisfying $c \le a$ and $d \le b$. Prove that $\frac{1}{r}$ is an integer.
Find all positive integers $a, b$ such that the numbers $\frac{a^2b + a}{a^2 + b}$ and $\frac{ab^2 + b}{b^2 - a}$ are integers.
Given a positive integer $n > 1$. Find the smallest integer of the form $\frac{n^a-n^b}{n^c-n^d}$ for all positive integers $a,b,c,d$.
a) Prove that for every positive integer n, there exist $a, b \in R - Z$ such that
the set $A_n = \{a - b, a^2 - b^2, a^3 - b^3,...,a^n - b^n\}$ contains only positive integers.
b) Let $a$ and $b$ be two real numbers such that the set $A = \{a^k - b^k | k \in N*\}$ contains only positive integers.
Prove that $a$ and $b$ are integers.
Given $\lambda^3 - 2\lambda^2- 1 = 0$ for some real $\lambda$ prove that $[\lambda[\lambda[\lambda n]]] - n$ is odd for any positive integer $n$ .
(I Voronovich)
Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.
Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.
Let $a, b$ and $c$ be pairwise relatively prime natural numbers. Find all possible values of $\frac{(a + b)(b + c)(c + a)}{abc}$ if known what it is integer.
Given the sequence $(t_n)$ defined as $t_0 = 0$, $t_1 = 6$, $t_{n + 2} = 14t_{n + 1} - t_n$.
Prove that for every number $n \ge 1$, $t_n$ is the area of a triangle whose lengths are all numbers integers.
Dang Hung Thang, University of Natural Sciences, Hanoi National University.