Found problems: 85335
Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$
Let $ABC$ be a right-angled triangle with $\angle ABC=90^\circ$, and let $D$ be on $AB$ such that $AD=2DB$. What is the maximum possible value of $\angle ACD$?
Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel.
[i]Advaith Avadhanam[/i]
Price of the book increased by $20\%$, and then decreased by $10\%$. How many percents should we decrease current price so we get a price which is $54\%$ percent of an original one?
Prove the inequality: $ctg \frac{a}{2}> 1 + ctg a$ for $0 <a <\frac{\pi}{2}$
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
Find the continuous functions $ f:\left[ 0,\frac{1}{3} \right] \longrightarrow (0,\infty ) $ that satisfy the functional relation
$$ 54\int_0^{1/3} f(x)dx +32\int_0^{1/3} \frac{dx}{\sqrt{x+f(x)}} =21. $$
[i]Cristinel Mortici[/i]
Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.
A small apartment building has four doors, with door numbers $1, 2, 3, 4.$ John has $2^4-1=15$ keys, label with of possible nonempty subsets of $\{1,2,3,4\}$, but he forgot which key is which. If an element on the key matches the door number, the key can open the door (e.g. key $\{1,2,4\}$ can open Door 4). He picks a key at random and tries to open Door 1, which fails, so he discards it. John then randomly picks one of his remaining 14 keys and tries to open Door 2, but it doesn't open, so he throws away that key as well. He then randomly selects one of the remaining 13 keys, and tests it on Door 3. What is the probability that it will open?
[i]Proposed by dantx5[/i]
How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ?
(A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above.
Note: $C^{m}_{n}$ stands for $\binom {m}{n}$
Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.
We’re playing a game with a sequence of $2008$ non-negative integers.
A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends.
Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.
How many 8-digit numbers are there such that exactly 7 digits are all different?
$\textbf{(A)}\ {{9}\choose{3}}^2 \cdot 6! \cdot 3 \qquad\textbf{(B)}\ {{8}\choose{3}}^2 \cdot 7! \qquad\textbf{(C)}\ {{8}\choose{3}}^2 \cdot 7! \cdot 3 \\ \qquad\textbf{(D)}\ {{7}\choose{3}}^2 \cdot 7! \qquad\textbf{(E)}\ {{9}\choose{4}}^2 \cdot 6! \cdot 8$
Let $\omega$ be the circumcircle of acute triangle $ABC$ where $\angle A<\angle B$ and $M,N$ be the midpoints of minor arcs $BC,AC$ of $\omega$ respectively. The line $PC$ is parallel to $MN$, intersecting $\omega$ at $P$ (different from $C$). Let $I$ be the incentre of $ABC$ and let $PI$ intersect $\omega$ again at the point $T$.
1) Prove that $MP\cdot MT=NP\cdot NT$;
2) Let $Q$ be an arbitrary point on minor arc $AB$ and $I,J$ be the incentres of triangles $AQC,BCQ$. Prove that $Q,I,J,T$ are concyclic.
Kelvin and Quinn are collecting trading cards; there are $6$ distinct cards that could appear in a pack. Each pack contains exactly one card, and each card is equally likely. Kelvin buys packs until he has at least one copy of every card, and then he stops buying packs. If Quinn is missing exactly one card, the probability that Kelvin has at least two copies of the card Quinn is missing is expressible as $\tfrac{m}{n}$ for coprime positive integers $m$ and $n$. Determine $m+n$.
a) Given an acute triangle $ABC(AB>AC).$ The circle $(O)$ with diameter $BC$ intersects $AB,AC$ at $F,E$, respectively. Let $H$ be the intersection point of $BE,CF,$ the line $AH$ intersects the line $BC$ at $D,$ the line $EF$ intersects the line $BC$ at $K.$ The line passing through $D$ and parallel to $EF$ intersects $AB,AC$ at $M,N,$ respectively.
Prove that: $M,O,N,K$ are on the same circle.
b) Given $\triangle ABC, \angle BAC=\angle BCA=30^{\circ}.$ $D,E,F$ are moving points on the side $AB,BC,CA$ such that: $\angle BFD=\angle BFE=60^{\circ}.$ Let $p,p_1$ be the perimeter of $\triangle ABC,\triangle DEF,$ respectively. Prove that: $p\le 2p_1.$
Square table $5 \times 5$ is filled with numbers in a following way.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8zLzQ0Y2M1NjdiNjQ3NjhlYTAwMWQ0MTg2ZjIwZWE4NzkwYzcwYWFkLnBuZw==&rn=dGFiZWxpY2EucG5n[/img]
We can change the table in a way we take two arbitrary numbers from the table and we decrease both of them with value of smaller of those two. Can we get to the table with all zeros?
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)
Given a quadrilateral $ABCD$ inscribed in $(O)$. Let $L, J$ be the [i]Lemoine[/i] point of $\triangle ABC$ and $\triangle ACD$.
Prove that: $AC, BD, LJ$ are concurrent.
Let $n$ be a positive integer. Given $n^2$ points in a unit square, prove that there exists a broken line of length $2n + 1$ that passes through all the points.
[i]Allen Yuan[/i]
Inside the triangle $ ABC $ a point $ P $ is given; find a point $ Q $ on the perimeter of this triangle such that the broken line $ APQ $ divides the triangle into two parts with equal areas.
Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant for all $i=0$ to $i=3$, where $[\mathcal{P}]$ denotes area of the polygon $\mathcal{P}$ and indices are taken modulo 4. What is the magnitude of the $z$-coordinate of $T_0$?
Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$.
$\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l}
\textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\
\textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\
\textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\
\textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\
\textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\
\textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\
\textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\
\textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23\\\\
\textbf{(Y) }24 \end{array}$
Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.
Two circles $\omega_1$ and $\omega_2$ are tangent to line $\ell$ at the points $A$ and $B$ respectively. In addition, $\omega_1$ and $\omega_2 $are externally tangent to each other at point $D$. Choose a point $E$ on the smaller arc $BD$ of circle $\omega_2$. Line $DE$ intersects circle $\omega_1$ again at point $C$. Prove that $BE \perp AC$.
(Yurii Biletskyi)