Found problems: 85335
The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is:
$ \textbf{(A)}\ 32 \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 7$
Without using a calculator, find out what is greater: $\sin 28^o$ or $tg21^o$?
Prove that all quadrilaterals $ABCD$ where $\angle B = \angle D = 90^o$, $|AB| = |BC|$ and $|AD| + |DC| = 1$, have the same area.
[img]https://1.bp.blogspot.com/-55lHuAKYEtI/XzRzDdRGDPI/AAAAAAAAMUk/n8lYt3fzFaAB410PQI4nMEz7cSSrfHEgQCLcBGAsYHQ/s0/2016%2Bmohr%2Bp3.png[/img]
Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$
$\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}$
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
What is the volume of a cube whose surface area is twice that of a cube with volume $ 1$?
$ \textbf{(A)}\ \sqrt{2} \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 2\sqrt{2} \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 8$
Define a convex quadrilateral $\mathcal{P}$ on the plane. In a turn, it is allowed to take some vertex of $\mathcal{P}$, move it perpendicular to the current diagonal of $\mathcal{P}$ not containing it, so long as it never crosses that diagonal. Initially $\mathcal{P}$ is a parallelogram and after several turns, it is similar but not congruent to its original shape. Show that $\mathcal P$ is a rhombus.
[i]Proposed by Evan Chang (squareman), USA[/i]
Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$.
Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$.
Time allowed for this problem was 1 hour.
Let $ABCD$ be a trapezoid with parallel sides $AB$ and $CD$, where $BC\neq DA$. A circle passing through $C$ and $D$ intersects $AC, AD, BC, BD$ again at $W, X, Y, Z$ respectively. Prove that $WZ, XY, AB$ are concurrent.
Let $X,Y$ be two points on the side $BC$ of triangle $ABC$ such that $2XY=BC$ ($X$ is between $B,Y$). Let $AA'$ be the diameter of the circumcirle of triangle $AXY$. Let $P$ be the point where $AX$ meets the perpendicular from $B$ to $BC$, and $Q$ be the point where $AY$ meets the perpendicular from $C$ to $BC$. Prove that the tangent line from $A'$ to the circumcircle of $AXY$ passes through the circumcenter of triangle $APQ$.
[i]Proposed by Iman Maghsoudi[/i]
Prove that there is a positive integer $M$ for which the following statement holds:
For all prime numbers $p$, there is an integer $n$ for which $\sqrt{p} \le n \le M\sqrt{p}$ and $p \mod n \le \frac{n}{2020}$ .
Note: Here, $p \mod n$ denotes the unique integer $r \in {0, 1, ..., n - 1}$ for which $n|p -r$. In other words, $p \mod n$ is the residue of $p$ upon division by $n$.
Let $k$ be a real number, such that the equation $kx^2 + k = 3x^2 + 2-2kx$ has two real solutions different. Determine all possible values of $k$, such that the sum of the roots of the equation is equal to the product of the roots of the equation increased by $k$.
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.
Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.
Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.
It is known that there are polyhedrons whose faces are more numbered than the vertices. Find the smallest number of triangular faces that such a polyhedron can have.
In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.
Define a a sequence $ {<{a_n}>}^{\infty}_{n\equal{}1}$ as follows
$ a_n\equal{}0$, if number of positive divisors of $ n$ is [i]odd[/i]
$ a_n\equal{}1$, if number of positive divisors of $ n$ is [i]even[/i]
(The positive divisors of $ n$ include $ 1$ as well as $ n$.)Let $ x\equal{}0.a_1a_2a_3........$ be the real number whose decimal expansion contains $ a_n$ in the $ n$-th place,$ n\geq1$.Determine,with proof,whether $ x$ is rational or irrational.
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$.
a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$.
b) Find the angle between the plane $a$ and the plane $BB_1C_1$.
Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$.
(Vyacheslav Yasinsky)
Two congruent equilateral triangles $\triangle ABC$ and $\triangle DEF$ lie on the same side of line $BC$ so that $B$, $C$, $E$, and $F$ are collinear as shown. A line intersects $\overline{AB}$, $\overline{AC}$, $\overline{DE}$, and $\overline{EF}$ at $W$, $X$, $Y$, and $Z$, respectively, such that $\tfrac{AW}{BW} = \tfrac29$ , $\tfrac{AX}{CX} = \tfrac56$ , and $\tfrac{DY}{EY} = \tfrac92$. The ratio $\tfrac{EZ}{FZ}$ can then be written as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[asy]
size(200);
defaultpen(linewidth(0.6));
real r = 3/11, s = 0.52, l = 33, d=5.5;
pair A = (l/2,l*sqrt(3)/2), B = origin, C = (l,0), D = (3*l/2+d,l*sqrt(3)/2), E = (l+d,0), F = (2*l+d,0);
pair W = r*B+(1-r)*A, X = s*C+(1-s)*A, Y = extension(W,X,D,E), Z = extension(W,X,E,F);
draw(E--D--F--B--A--C^^W--Z);
dot("$A$",A,N);
dot("$B$",B,S);
dot("$C$",C,S);
dot("$D$",D,N);
dot("$E$",E,S);
dot("$F$",F,S);
dot("$W$",W,0.6*NW);
dot("$X$",X,0.8*NE);
dot("$Y$",Y,dir(100));
dot("$Z$",Z,dir(70));
[/asy]
Two of the numbers $a+b, a-b, ab, a/b$ are positive, the other two are negative. Find the sign of $b$
In trapezoid $ABCD$ shown, $AD$ is parallel to $BC$, and $AB = 6$, $BC = 7$, $CD = 8$; $AD = 17$. If sides $AB$ and $CD$ are extended to meet at $E$, find the resulting angle at $E$ (in degrees).
[img]https://cdn.artofproblemsolving.com/attachments/c/c/987dde1d841a7b976b01f597511e5c1e828e5a.png[/img]
$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?
Given a parallelepiped $P$, let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$, let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$. Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4} L_P t^2 + \frac{4\pi}{3} t^3$.