Found problems: 85335
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.
$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$
$(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$.
.
Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at point $X$ inside the square. How far is $X$ from side $CD$?
$\textbf{(A) }\frac{1}{2}s(\sqrt{3}+4)\qquad\textbf{(B) }\frac{1}{2}s\sqrt{3}\qquad\textbf{(C) }\frac{1}{2}s(1+\sqrt{3})\qquad$
$\textbf{(D) }\frac{1}{2}s(\sqrt{3}-1)\qquad \textbf{(E) }\frac{1}{2}s(2-\sqrt{3})$
A nine-digit telephone number [i]abcdefghi [/i] is called [i]memorizable [/i] if the sequence of four initial digits [i]abcd [/i] is repeated in the sequence of the final five digits [i]efghi[/i]. How many [i]memorizable [/i] numbers of nine digits exist?
Let \(n\) be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.
Consider a convex polygon $P$ with $n$ sides and perimeter $P_0$. Let the polygon $Q$, whose vertices are the midpoints of the sides of $P$, have perimeter $P_1$. Prove that $P_1 \geq \frac{P_0}{2}$.
Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses. Prove that there are infinitely many initial situations in which the second player can win no matter how his opponent plays.
$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s.
[i]Proposed by Aryan Tajmir[/i]
Let $n \geq 2$ be a fixed positive integer and let $a_{0},a_{1},...,a_{n-1}$ be real numbers. Assume that all of the roots of the polynomial $P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0}$ are strictly positive real numbers. Determine the smallest possible value of $\frac{a_{n-1}^{2}}{a_{n-2}}$ over all such polynomials.
[i]Proposed by Nikola Velov[/i]
Ankit, Bill, Charlie, Druv, and Ed are playing a game in which they go around shouting numbers in that order. Ankit starts by shouting the number $1$. Bill adds a number that is a factor of the number of letters in his name to Ankit’s number and shouts the result. Charlie does the same with Bill’s number, and so on (once Ed shouts a number, Ankit does the same procedure to Ed’s number, and the game goes on). What is the sum of all possible numbers that can be the $23$rd shout?
In a given tedrahedron $ ABCD$ let $ K$ and $ L$ be the centres of edges $ AB$ and $ CD$ respectively. Prove that every plane that contains the line $ KL$ divides the tedrahedron into two parts of equal volume.
Find all real numbers $x, y, z$ that satisfy the following system
$$\sqrt{x^3 - y} = z - 1$$
$$\sqrt{y^3 - z} = x - 1$$
$$\sqrt{z^3 - x} = y - 1$$
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.
For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.
In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.
Prove that among $39$ sequential natural numbers there always is a number with the sum of its digits divisible by $11$.
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),
$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.
[i]O. Musin[/i]
What is the remainder when $13^{16} + 17^{12}$ is divided by $221$?
In a tetrahedron $ABCD, E$ and $F$ are the midpoints of the medians from $A$ and $D$. Find the ratio of the volumes of tetrahedra $BCEF$ and $ABCD$.
Note: Median in a tetrahedron connects a vertex and the centroid of the opposite side.
The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?
Consider this histogram of the scores for $81$ students taking a test:
[asy]
unitsize(12);
draw((0,0)--(26,0));
draw((1,1)--(25,1));
draw((3,2)--(25,2));
draw((5,3)--(23,3));
draw((5,4)--(21,4));
draw((7,5)--(21,5));
draw((9,6)--(21,6));
draw((11,7)--(19,7));
draw((11,8)--(19,8));
draw((11,9)--(19,9));
draw((11,10)--(19,10));
draw((13,11)--(19,11));
draw((13,12)--(19,12));
draw((13,13)--(17,13));
draw((13,14)--(17,14));
draw((15,15)--(17,15));
draw((15,16)--(17,16));
draw((1,0)--(1,1));
draw((3,0)--(3,2));
draw((5,0)--(5,4));
draw((7,0)--(7,5));
draw((9,0)--(9,6));
draw((11,0)--(11,10));
draw((13,0)--(13,14));
draw((15,0)--(15,16));
draw((17,0)--(17,16));
draw((19,0)--(19,12));
draw((21,0)--(21,6));
draw((23,0)--(23,3));
draw((25,0)--(25,2));
for (int a = 1; a < 13; ++a)
{
draw((2*a,-.25)--(2*a,.25));
}
label("$40$",(2,-.25),S);
label("$45$",(4,-.25),S);
label("$50$",(6,-.25),S);
label("$55$",(8,-.25),S);
label("$60$",(10,-.25),S);
label("$65$",(12,-.25),S);
label("$70$",(14,-.25),S);
label("$75$",(16,-.25),S);
label("$80$",(18,-.25),S);
label("$85$",(20,-.25),S);
label("$90$",(22,-.25),S);
label("$95$",(24,-.25),S);
label("$1$",(2,1),N);
label("$2$",(4,2),N);
label("$4$",(6,4),N);
label("$5$",(8,5),N);
label("$6$",(10,6),N);
label("$10$",(12,10),N);
label("$14$",(14,14),N);
label("$16$",(16,16),N);
label("$12$",(18,12),N);
label("$6$",(20,6),N);
label("$3$",(22,3),N);
label("$2$",(24,2),N);
label("Number",(4,8),N);
label("of Students",(4,7),N);
label("$\textbf{STUDENT TEST SCORES}$",(14,18),N);
[/asy]
The median is in the interval labeled
$\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80$
$f(n)$ denotes the largest integer $k$ such that that $2^k|n$.
$2006$ integers $a_i$ are such that $a_1<a_2<...<a_{2016}$.
Is it possible to find integers $k$ where $1 \le k\le 2006$ and $f(a_i-a_j)\ne k$ for every $1 \le j \le i \le 2006$ ?
It is possible to place an even number of pears in a row such that the masses of any two neighbouring pears differ by at most $1$ gram. Prove that it is then possible to put the pears two in a bag and place the bags in a row such that the masses of any two neighbouring bags differ by at most $1$ gram.
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?