Found problems: 85335
In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$.
An acute triangle $ABC$ is inscribed in circle $\omega$ centered at $O$. Line $BO$ and side $AC$ meet at $B_1$. Line $CO$ and side $AB$ meet at $C_1$. Line $B_1C_1$ meets circle $\omega$ at $P$ and $Q$. If $AP = AQ$, prove that $AB = AC$.
Let $n \ge 3$ be a positive integer, and let $S$ be a set of $n$ distinct points in the plane. Call an unordered pair of distinct points ${A,B}$ [i]tasty[/i] if there exists a circle passing through $A$ and $B$ not passing through or containing any other point in $S$. Find the maximum number of tasty pairs over all possible sets $S$ of $n$ points.
[i]Tiger Zhang[/i]
Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.
Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.
Let $n>2$ be a given positive integer. There are $n$ guests at Georg's bachelor party and each guest is friends with at least one other guest. Georg organizes a party game among the guests. Each guest receives a jug of water such that there are no two guests with the same amount of water in their jugs. All guests now proceed simultaneously as follows. Every guest takes one cup for each of his friends at the party and distributes all the water from his jug evenly in the cups. He then passes a cup to each of his friends. Each guest having received a cup of water from each of his friends pours the water he has received into his jug. What is the smallest possible number of guests that do not have the same amount of water as they started with?
Initially one have the number $0$ in each cell of the table $29 \times 29$. A [i]moviment[/i] is when one choose a sub-table $5 \times 5$ and add $+1$ for every cell of this sub-table. Find the maximum value of $n$, where after $1000$ [i]moviments[/i], there are $4$ cells such that your centers are vertices of a square and the sum of this $4$ cells is at least $n$.
[b]Note:[/b] A square does not, necessarily, have your sides parallel with the sides of the table.
Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that:
$$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$
In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy]
There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.
Let $x,y,z,t$ be real numbers such that, $1 \leq x \leq y \leq z \leq t \leq 100$. Find minimum value of
$$\frac{x}{y}+\frac{z}{t}$$
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
[asy]
size(7cm);
draw((-8,27)--(72,27));
draw((16,0)--(16,35));
draw((40,0)--(40,35));
label("12", (28,3));
draw((25,6.5)--(25,12)--(31,12)--(31,6.5)--cycle);
draw((25,5.5)--(31,5.5));
label("18", (56,3));
draw((53,6.5)--(53,12)--(59,12)--(59,6.5)--cycle);
draw((53,5.5)--(59,5.5));
draw((53,5.5)--(59,5.5));
label("20", (28,18));
label("15", (28,24));
draw((25,21)--(31,21));
label("10", (56,18));
label("10", (56,24));
draw((53,21)--(59,21));
label("First Half", (28,31));
label("Second Half", (56,31));
label("Candace", (2.35,6));
label("Steph", (0,21));
[/asy]
$\textbf{(A)} ~7\qquad\textbf{(B)} ~8\qquad\textbf{(C)} ~9\qquad\textbf{(D)} ~10\qquad\textbf{(E)} ~11$
Find minimal natural $n$ for which there exist integers $a_1, a_2,\ldots, a_n$ such that quadratic trinom
$$x^2-2(a_1+a_2+\cdots+a_n)^2x+(a_1^4+a_2^4+\cdots+a_n^4+1)$$
has at least one integral root.
The circumferences $C_1$ and $C_2$ cut each other at different points $A$ and $K$. The common tangent to $C_1$ and $C_2$ nearer to $K$ touches $C_1$ at $B$ and $C_2$ at $C$. Let $P$ be the foot of the perpendicular from $B$ to $AC$, and let $Q$ be the foot of the perpendicular from $C$ to $AB$. If $E$ and $F$ are the symmetric points of $K$ with respect to the lines $PQ$ and $BC$, respectively, prove that $A, E$ and $F$ are collinear.
We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point.
b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.
Points $C_1, A_1$ and $B_1$ are marked on the sides $AB, BC$ and $CA$ of a triangle $ABC$ so that the segments $AA_1, BB_1$, and $CC_1$ are concurrent (see the fig.). It is known that the area of the white part of the triangle $ABC$ is equal to the area of its black part. Prove that at least one of the segments $AA_1, BB_1, CC_1$ is a median of the triangle $ABC$.
[img]https://1.bp.blogspot.com/-nVVhqdRdf0s/X-WVmt_gyqI/AAAAAAAAM40/943sCRGyCPwT-vqIilTCtXOXHByRLIvPwCLcBGAsYHQ/s0/2014%2Bbelarus%2B11.6.png[/img]
The graphs of $ y \equal{} \minus{}|x \minus{} a| \plus{} b$ and $ y \equal{} |x \minus{} c| \plus{} d$ intersect at points $ (2,5)$ and $ (8,3)$. Find $ a \plus{} c$.
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 13\qquad
\textbf{(E)}\ 18$
Define two sequences $x_n, y_n$ for $n = 1, 2, \ldots$ by \[x_n = \left(\sum^n_{k=0} \binom{2n}{2k}49^k 48^{n-k} \right) -1, \quad \text{and} \quad y_n = \sum^{n-1}_{k=0} \binom{2n}{2k + 1} 49^k 48^{n-k}\] Prove there is a positive integer $m$ for which for every integer $n > m,$ the greatest common factor of $x_n$ and $y_n$ is more than $10^{2024}.$
Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:
\begin{align*}
a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\
a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\
a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\
&\vdots & &\vdots \\
a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1}
\end{align*}
Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.
Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$
[i]Michael Ren and Ankan Bhattacharya, USA[/i]
The audience chooses two of twenty-nine cards, numbered from $1$ to $29$ respectively. The assistant of a magician chooses two of the remaining twenty-seven cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in an arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.
Find $PB$, given that $PA = 15$, $PC = 20$, $PD = 7$, and $ABCD$ is a square.
[img]https://cdn.artofproblemsolving.com/attachments/7/a/cc5bf99986fea1cd75e57fe1117a4d04d3eae3.png[/img]
(i) For positive integers $a<b$, let $M(a,b)=\frac{\Sigma^{b}_{k=a}\sqrt{k^2+3k+3}}{b-a+1}$.
Calculate $[M(a,b)]$
(ii) Calculate $N(a,b)=\frac{\Sigma^{b}_{k=a}[\sqrt{k^2+3k+3}]}{b-a+1}$.