Found problems: 85335
The value of the expression $ \frac{(304)^5}{(29.7)(399)^4}$ is closest to
\[ \textbf{(A)}\ .003 \qquad
\textbf{(B)}\ .03 \qquad
\textbf{(C)}\ .3 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 30
\]
Prove that for every positive integer $m$, every prime $p$ and every positive integer $j \le p^{m-1}$,
$p^m$ divides $${p^m \choose p^j }- {p^{m-1} \choose j}$$
Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.
In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$, 2, 3, and 4 ($A_5 = A_1$). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$.
[asy]
unitsize(1 cm);
pair O;
pair[] A, B;
O = (0,0);
A[1] = (0.5,-3);
A[2] = (2,0);
A[3] = (-0.2,0.5);
A[4] = (-1,0);
B[1] = A[2] - A[1];
B[2] = A[3] - A[2];
B[3] = A[4] - A[3];
B[4] = A[1] - A[4];
draw(A[1]--A[2]--A[3]--A[4]--cycle);
draw(B[1]--B[2]--B[3]--B[4]--cycle);
draw(O--B[1]);
draw(O--B[2]);
draw(O--B[3]);
draw(O--B[4]);
label("$A_1$", A[1], S);
label("$A_2$", A[2], E);
label("$A_3$", A[3], N);
label("$A_4$", A[4], W);
label("$B_1$", B[1], NE);
label("$B_2$", B[2], W);
label("$B_3$", B[3], SW);
label("$B_4$", B[4], S);
label("$O$", O, E);
[/asy]
Let $n$ be a positive integer. In an $n \times n$ board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place $n$ queens on the board such that no two threaten each other when:
$a)\:$ $n=14$.
$b)\:$ $n=15$.
Elisenda has a piece of paper in the shape of a triangle with vertices $A, B,$ and $C$ such that $AB = 42.$ She chooses a point $D$ on segment $AC,$ and she folds the paper along line $BD$ so that $A$ lands at a point $E$ on segment $BC.$ Then, she folds the paper along line $DE.$ When she does this, $B$ lands at the midpoint of segment $DC.$ Compute the perimeter of the original unfolded triangle.
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]
Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.
There are $28$ students who have to be separated into two groups such that the number of students in each group
is a multiple of $4$. The number of ways to split them into the groups can be written as
$$\sum_{k \ge 0} 2^k a_k = a_0 +2a_1 +4a_2 +...$$
where each $a_i$ is either $0$ or $1$. Find the value of
$$\sum_{k \ge 0} ka_k = 0+ a_1 +2a_2 +3a3_ +....$$
Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t.
\[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]
Prove that for all sufficiently large positive integers $d{}$, at least $99\%$ of the polynomials of the form \[\sum_{i\leqslant d}\sum_{j\leqslant d}\pm x^iy^j\]are irreducible over the integers.
Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that
\[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]
Find all pairs of $p,q$ prime numbers that satisfy the equation
$$p(p^4+p^2+10q)=q(q^2+3)$$
Calculates the sum of all numbers that can be formed using each of the odd digits once, that is, the numbers $13579$, $13597$, ..., $97531$.
Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.
Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number n when read from left to right. Compute the expected value of $n$.
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that:
$$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$
For a quadratic trinomial $f(x)$ and the different numbers $a$ and $b$ it is known that $f(a)=b$ and $f(b)=a$. We call such $a$ and $b$ [i]conjugate[/i] for $f(x)$. Prove that $f(x)$ has no other [i]conjugate[/i] numbers.
In a certain store, all pencils cost the same amount of money. If three pencils can be bought for six dollars, what is the price of two pencils?
$\textbf{(A) }\$ 3\qquad\textbf{(B) }\$ 3.5\qquad\textbf{(C) }\$ 4\qquad\textbf{(D) }\$4.5\qquad\textbf{(E) }\$ 5$
Prove that if $ n $ is a non-negative integer, then number $$
2^{n+2} + 3^{2n+1}$$
is divisible by $7$.
If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.
Let $\{ a_n\}$ be a series consisting of positive integers such that $n^2 \mid \sum_{i=1}^{n}{a_i}$ and $a_n\leq (n+2016)^2$ for all $n\geq 2016$.
Define $b_n=a_{n+1}-a_n$. Prove that the series $\{ b_n\}$ is eventually constant.
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.
Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.
In circle $\omega$, two perpendicular chords intersect at a point $P$. The two chords have midpoints $M_1$ and $M_2$ respectively, such that $PM_1=15$ and $PM_2=20$. Line $M_1M_2$ intersects $\omega$ at points $A$ and $B$, with $M_1$ between $A$ and $M_2$. Compute the largest possible value of $BM_2-AM_1$.