Found problems: 85335
Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$.
Show that at least one between $a - 1, b - 1, c -1$ is composite number.
Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$
How many ways are there to completely fill a $3 \times 3$ grid of unit squares with the letters $B, M$, and $T$, assigning exactly one of the three letters to each of the squares, such that no $2$ adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.
Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$
What is the greatest number of balls of radius $1/2$ that can be placed within a rectangular box of size $10 \times 10 \times 1 \ ?$
Let $N$ be a natural number. Find (with prove) the number of solutions in the segment $[1,N]$ of the equation $x^2-[x^2]=(x-[x])^2$, where $[x]$ means the floor function of $x$.
We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$
Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
$\triangle ABC$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $\omega_1$ be the incircle of $\triangle ABM$ and $\omega_2$ be the incircle of $\triangle BCM$. The line externally tangent to $\omega_1$ and $\omega_2$ that is not $AC$ intersects $AB$ and $BC$ at $X$ and $Y$, respectively. If the area of $\triangle BXY$ can be expressed as $\frac{m}{n}$, compute is $m+n$.
[i]Proposed by Alex Li[/i]
Suppose line $\ell$ and four points $A,B,C,D$ lies on $\ell$. Suppose that circles $\omega_1 , \omega_2$ passes through $A,B$ and circles $\omega'_1 , \omega'_2$ passes through $C,D$. If $\omega_1 \perp \omega'_1$ and $\omega_2 \perp \omega'_2$ then prove that lines $O_1O'_2 , O_2O'_1 , \ell $ are concurrent where $O_1,O_2,O'_1,O'_2$ are center of $\omega_1 , \omega_2 , \omega'_1 , \omega'_2$.
Does there exist a nonzero polynomial $P(x)$ with integer coefficients satisfying both of the following conditions?
[list]
[*]$P(x)$ has no rational root;
[*]For every positive integer $n$, there exists an integer $m$ such that $n$ divides $P(m)$.
[/list]
Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$.
a) Find the angles $\angle B$ and $\angle C$
b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.
Find all the integers $x$ in $[20, 50]$ such that $6x+5\equiv 19 \mod 10$, that is, $10$ divides $(6x+15)+19$.
For each subset $A$ of $\{1,2, \dots, n \} $, let $M_A$ be the difference between the largest of the elements of $A$ and the smallest of the elements of $A $. Finds the sum of all values of $M_A$ when all possible subsets $A$ of $\{1,2, \dots, n \} $ are considered.
Construct a triangle given the radii of the excircles.
The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$.
Harold has $3$ red checkers and $3$ black checkers. Find the number of distinct ways that Harold can place these checkers in stacks. Two ways of stacking checkers are the same if each stack of the rst way matches a corresponding stack in the second way in both size and color arrangement. So, for example, the $3$ stack arrangement $RBR, BR, B$ is distinct from $RBR, RB, B$, but the $4$ stack arrangement $RB, BR, B, R$ is the same as $B, BR, R, RB$.
Kevin is in first grade, so his teacher asks him to calculate $20+1\cdot 6+k$, where $k$ is a real number revealed to Kevin. However, since Kevin is rude to his Aunt Sally, he instead calculates $(20+1)\cdot (6+k)$. Surprisingly, Kevin gets the correct answer! Assuming Kevin did his computations correctly, what was his answer?
[i]Proposed by James Lin[/i]
Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $P$. The circumscribed circles of triangles $APD$ and $BPC$ intersect the line $AB$ at points $E, F$ correspondingly. $Q$ and $R$ are the projections of $P$ onto the lines $FC, DE$ correspondingly. Show that $AB \parallel QR$.
[i](Proposed by Mykhailo Shtandenko)[/i]
If, instead, the graph is a graph of VELOCITY vs. TIME, then the squirrel has the greatest speed at what time(s) or during what time interval(s)?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
[b]a)[/b] Prove that any multiple of $ 6 $ is the sum of four cubes.
[b]b)[/b] Show that any integer is the sum of five cubes.
Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and
$ x_1\plus{}...\plus{}x_n \equiv k$ mod 4
a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$.
(b) Prove that
$ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$
for all positive integers $ n$.
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $ \$1.43$. Some of the $ 30$ sixth graders each bought a pencil, and they paid a total of $ \$1.95$. How many more sixth graders than seventh graders bought a pencil?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer.
a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice.
b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?