Found problems: 85335
In right triangle $ABC$, $\angle ACB = 90^{\circ}$ and $\tan A > \sqrt{2}$. $M$ is the midpoint of $AB$, $P$ is the foot of the altitude from $C$, and $N$ is the midpoint of $CP$. Line $AB$ meets the circumcircle of $CNB$ again at $Q$. $R$ lies on line $BC$ such that $QR$ and $CP$ are parallel, $S$ lies on ray $CA$ past $A$ such that $BR = RS$, and $V$ lies on segment $SP$ such that $AV = VP$. Line $SP$ meets the circumcircle of $CPB$ again at $T$. $W$ lies on ray $VA$ past $A$ such that $2AW = ST$, and $O$ is the circumcenter of $SPM$. Prove that lines $OM$ and $BW$ are perpendicular.
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$.
(Proposed by Stephan Wagner, Stellenbosch University)
Four distinct points are arranged in a plane so that the segments connecting them has lengths $a,a,a,a,2a,$ and $b$. What is the ratio of $b$ to $a$?
$ \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi $
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.
Prove that line $AO$ passes through the midpoint of segment $BE$.
Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying:
\[\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c \]
then for every $i,j,k$ such that $1\le 1<j<k\le 17$, we have that $x_i,x_j,x_k$ are sides of a triangle.
Jody has $6$ distinguishable balls and $6$ distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.
Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$
If $\sin x+\cos x=1/5$ and $0\le x<\pi$, then $\tan x$ is
$\textbf{(A) }-\frac{4}{3}\qquad\textbf{(B) }-\frac{3}{4}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{4}{3}\qquad$
$\textbf{(E) }\text{not completely determined by the given information}$
In convex hexagon $ABCDEF$, all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B$, $\angle C$, $\angle E$, and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$. Find $AB$.
A regular octagon $ ABCDEFGH$ has sides of length two. Find the area of $ \triangle{ADG}$.
$ \textbf{(A)}\ 4 \plus{} 2 \sqrt{2} \qquad
\textbf{(B)}\ 6 \plus{} \sqrt{2} \qquad
\textbf{(C)}\ 4 \plus{} 3 \sqrt{2} \qquad
\textbf{(D)}\ 3 \plus{} 4 \sqrt{2} \qquad
\textbf{(E)}\ 8 \plus{} \sqrt{2}$
Which regular polygon can be obtained (and how) by cutting a cube with a plane ?
The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]
Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.
Solve in real numbers the equation $$\log_7 (6^x+1)=\log_6(7^x-1).$$
[i]Mathematical Gazette[/i]
Acute triangle $ABC$ satisfies $\angle A > \angle C$. Let $D, E, F$ be the points that the triangle's incircle intersects with $BC, CA, AB$, respectively, and $P$ some point on $AF$ different from $F$. The angle bisector of $\angle ABC$ meets $PQR$'s circumcircle $O$ at $L, R$. $L$ is the point closer to $B$ than $R$. $O$ meets $DF, DR$ at point $Q(\neq F, L), S(\neq R)$ respectively, and $PS$ hits segment $BC$ at $T$. Show that $T, Q, L$ are collinear.
Prove that $\sqrt 2 +\sqrt 3 +\sqrt{1990}$ is irrational.
[b]p1. [/b] Evaluate $S$.
$$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$
[b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?
[b]p3.[/b] Given that $$(a + b) + (b + c) + (c + a) = 18$$
$$\frac{1}{a + b}+\frac{1}{b + c}+ \frac{1}{c + a}=\frac59,$$
determine $$\frac{c}{a + b}+\frac{a}{b + c}+\frac{b}{c + a}.$$
[b]p4.[/b] Find all primes $p$ such that $2^{p+1} + p^3 - p^2 - p$ is prime.
[b]p5.[/b] In right triangle $ABC$ with the right angle at $A$, $AF$ is the median, $AH$ is the altitude, and $AE$ is the angle bisector. If $\angle EAF = 30^o$ , find $\angle BAH$ in degrees.
[b]p6.[/b] For which integers $a$ does the equation $(1 - a)(a - x)(x- 1) = ax$ not have two distinct real roots of $x$?
[b]p7. [/b]Given that $a^2 + b^2 - ab - b +\frac13 = 0$, solve for all $(a, b)$.
[b]p8. [/b] Point $E$ is on side $\overline{AB}$ of the unit square $ABCD$. $F$ is chosen on $\overline{BC}$ so that $AE = BF$, and $G$ is the intersection of $\overline{DE}$ and $\overline{AF}$. As the location of $E$ varies along side $\overline{AB}$, what is the minimum length of $\overline{BG}$?
[b]p9.[/b] Sam and Susan are taking turns shooting a basketball. Sam goes first and has probability $P$ of missing any shot, while Susan has probability $P$ of making any shot. What must $P$ be so that Susan has a $50\%$ chance of making the first shot?
[b]p10.[/b] Quadrilateral $ABCD$ has $AB = BC = CD = 7$, $AD = 13$, $\angle BCD = 2\angle DAB$, and $\angle ABC = 2\angle CDA$. Find its area.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 11 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 23 \qquad
\textbf{(E)}\ 77
$
Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following:
There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements,
$$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$
where $S(n)$ denotes sum of digits of decimal representation of $n$.
In a round-robin tournament with $n$ players $P_1$, $P_2$, $\ldots$, $P_n$ (where $n > 1$), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $w_r$ and $l_r$ be the number of games won and lost, respectively, by $P_r$. Show that
\[
\sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2.
\]
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]
Let $n > 1$ be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers $x$ and $y$ satisfying $$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}$$ is that $n$ has a prime divisor of the form $4k - 1$.
The integers $a_1,a_2,...,a_{2012}$ are given. Exactly $29$ of them are divisible by $3$. Prove that the sum $a_1^2+a_2^2+...+a_{2012}^2$ is divisible by $3$.
Let $p$ and $k$ be positive integers such that $p$ is prime and $k>1$. Prove that there is at most one pair $(x,y)$ of positive integers such that
$x^k+px=y^k$.
The points $A_1,B_1,C_1$ lie on the sides $BC,CA$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $O_A,O_B$ and $O_C$ be the circumcentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the incentre of triangle $O_AO_BO_C$ is the incentre of triangle $ABC$ too.