Found problems: 85335
Most enzymes are a type of
$ \textbf{(A) } \text{Carbohydrate} \qquad\textbf{(B) } \text{Lipid} \qquad\textbf{(C) } \text{Nucleic Acid} \qquad\textbf{(D) } \text{Protein} \qquad $
Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?
$\omega$ is a circle inside angle $\measuredangle BAC$ and it is tangent to sides of this angle at $B,C$.An arbitrary line $ \ell $ intersects with $AB,AC$ at $K,L$,respectively and intersect with $\omega$ at $P,Q$.Points $S,T$ are on $BC$ such that $KS \parallel AC$ and $TL \parallel AB$.Prove that $P,Q,S,T$ are concyclic.(I.Bogdanov,P.Kozhevnikov)
Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices.
a) Find all possible values of pair $(A,B).$
b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.
Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that:
\[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\]
where R is the radius of circumcircle of $\triangle ABC$.
[b]Problem 3.[/b] Two points $M$ and $N$ are chosen inside a non-equilateral triangle $ABC$ such that $\angle BAM=\angle CAN$, $\angle ABM=\angle CBN$ and \[AM\cdot AN\cdot BC=BM\cdot BN\cdot CA=CM\cdot CN\cdot AB=k\] for some real $k$. Prove that:
[b]a)[/b] We have $3k=AB\cdot BC\cdot CA$.
[b]b)[/b] The midpoint of $MN$ is the medicenter of $\triangle ABC$.
[i]Remark.[/i] The [b]medicenter[/b] of a triangle is the intersection point of the three medians:
If $A_{1}$ is midpoint of $BC$, $B_{1}$ of $AC$ and $C_{1}$ of $AB$, then $AA_{1}\cap BB_{1}\cap CC_{1}= G$, and $G$ is called medicenter of triangle $ABC$.
[i] Nikolai Nikolov[/i]
The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$, and on it the points $M$ and $N$ are taken such that $BM = NC = k$, with $k < a/2$. Assuming that (only) the data $a$ and $k$ are known, calculate:
a) The value of the sum of the squares of the lengths $AM$ and $AN$.
b) The ratio of the areas of triangles $ABC$ and $AMN$.
c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$.
$\{a_n\}$ is a positive integer sequence such that $ a_{i+2} = a_{i+1} + a_{i} (i \ge 1) $. For positive integer $n$, define $\{b_n\}$ as
\[ b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i } \]
Prove that $b_n$ is positive integer, and find the general form of $b_n$.
Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution][i]Solution[/i]. $\boxed{186}$
We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]
Consider a point $ B $ on a segment $ AC. $ Find the locus of the points $ M $ that have the property that the circumcircles of $ ABM $ and $ BCM $ have equal radii.
[i]Nicolae Soare[/i]
For a triangle $ ABC $, let $ B_1 ,C_1 $ be the excenters of $ B, C $. Line $B_1 C_1 $ meets with the circumcircle of $ \triangle ABC $ at point $ D (\ne A) $. $ E $ is the point which satisfies $ B_1 E \bot CA $ and $ C_1 E \bot AB $. Let $ w $ be the circumcircle of $ \triangle ADE $. The tangent to the circle $ w $ at $ D $ meets $ AE $ at $ F $. $ G , H $ are the points on $ AE, w $ such that $ DGH \bot AE $. The circumcircle of $ \triangle HGF $ meets $ w $ at point $ I ( \ne H ) $, and $ J $ be the foot of perpendicular from $ D $ to $ AH $. Prove that $ AI $ passes the midpoint of $ DJ $.
Prove that $102^{1991} + 103^{1991}$ is not a proper power of an integer.
Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.
Let $n$ be a positive integer. A grid of dimensions $n \times n$ is divided in $n^2$ $1 \times 1$ squares. Every segment of length $1$ (side of a square) from this grid is coloured in blue or red. The number of red segments is not greater than $n^2$. Find all positive integers $n$, for which the grid always will cointain at least one $1 \times 1$ square which has at least three blue sides.
We have to divide a cube onto $k$ non-overlapping tetrahedrons. For what smallest $k$ is it possible?
Solve the equations :
$$\begin{cases} a + b + c = 0 \\ a^2 + b^2 + c^2 = 1\\a^3 + b^3 +c^3 = 4abc \end{cases}$$ for $ a,b,$ and $c. $
Five elders are sitting around a large bonfire. They know that Oluf will put a hat of one of four colours (red, green, blue or yellow) on each elder’s head, and after a short time for silent reflection each elder will have to write down one of the four colours on a piece of paper. Each elder will only be able to see the colour of their two neighbours’ hats, not that of their own nor that of the remaining two elders’ hats, and they also cannot communicate after Oluf starts putting the hats on.
Show that the elders can devise a strategy ahead of time so that at most two elders will end up writing down the colour of their own hat
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$
(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
Kiana has two older twin brothers. The product of their ages is $ 128$. What is the sum of their three ages?
$ \textbf{(A)}\ 10\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 16\qquad
\textbf{(D)}\ 18\qquad
\textbf{(E)}\ 24$
Lily has an unfair coin that has $\tfrac23$ probability of showing heads and $\tfrac13$ probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails?
$\textbf{(A) }0\qquad\textbf{(B) }1/9\qquad\textbf{(C) }2/9\qquad\textbf{(D) }4/9\qquad\textbf{(E) }1$
A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
$ \textbf{(A)}\ 36\pi\minus{}34 \qquad
\textbf{(B)}\ 30\pi \minus{} 15 \qquad
\textbf{(C)}\ 36\pi \minus{} 33 \qquad$
$ \textbf{(D)}\ 35\pi \minus{} 9\sqrt3 \qquad
\textbf{(E)}\ 30\pi \minus{} 9\sqrt3$
A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements of $A$. $A-B$ is the set of elements that are in $A$ but not $B$. $\text{min}(x,y)$ is the number that is not larger than the other.)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots,n\}$
(ii): For all $1 \le i < j \le m$, $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
The length $l$ of a tangent, drawn from a point $A$ to a circle, is $\frac43$ of the radius $r$. The (shortest) distance from $A$ to the circle is:
$ \textbf{(A)}\ \frac{1}{2}r \qquad\textbf{(B)}\ r\qquad\textbf{(C)}\ \frac{1}{2}l\qquad\textbf{(D)}\ \frac23l \qquad\textbf{(E)}\ \text{a value between r and l.} $
On the plane there is an image of a circle with a marked center. Let an arbitrary angle be drawn on this plane. Using one ruler, construct the bisector of this angle.