Found problems: 85335
Let $S_n=\sum_{k=1}^n (k^5+k^7).$ Let the prime factorization of $\text{gcd}(S_{2020},S_{6060})$ be $p_1^{k_1}\cdot p_2^{k_2}\cdots p_i^{k_i}$. Compute $p_1+p_2+\cdots +p_i+k_1+k_2+\cdots + k_i $.
Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar.
[i]Fedir Yudin [/i]
The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star.
[img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]
Given a natural number $n\geqslant 2$, find the smallest possible number of edges in a graph that has the following property: for any coloring of the vertices of the graph in $n{}$ colors, there is a vertex that has at least two neighbors of the same color as itself.
$n, k$ are positive integers. $A_0$ is the set $\{1, 2, ... , n\}$. $A_i$ is a randomly chosen subset of $A_{i-1}$ (with each subset having equal probability). Show that the expected number of elements of $A_k$ is $\dfrac{n}{2^k}$
In quadrilateral $ABCD$, $E$ and $F$ are midpoints of $AB$ and $CD$, and $G$ is the intersection of $AD$ with $BC$. $P$ is a point within the quadrilateral, such that $PA=PB$, $PC=PD$, and $\angle APB+\angle CPD=180^{\circ}$. Prove that $PG$ and $EF$ are parallel.
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.
[i]Jakob Jurij Snoj, Slovenia[/i]
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.
What is the number of $4$ digit natural numbers such that the sum of digits is even?
(A)$4999$
(B)$5000$
(C)$5050$
(D)$4500$
Prove that the equation \[ x^{2008}\plus{} 2008!\equal{} 21^{y}\] doesn't have solutions in integers.
Juli is a mathematician and devised an algorithm to find a husband. The strategy is:
• Start interviewing a maximum of $1000$ prospective husbands. Assign a ranking $r$ to each person that is a positive integer. No two prospects will have same the rank $r$.
• Reject the first $k$ men and let $H$ be highest rank of these $k$ men.
• After rejecting the first $k$ men, select the next prospect with a rank greater than $H$ and then stop the search immediately. If no candidate is selected after $999$ interviews, the $1000th$ person is selected.
Juli wants to find the value of $k$ for which she has the highest probability of choosing the highest ranking prospect among all $1000$ candidates without having to interview all $1000$ prospects.
[b](a)[/b] (6 points:) What is the probability that the highest ranking prospect among all $1000$ prospects is the $(m + 1)th$ prospect?
[b](b)[/b] (6 points:) Assume the highest ranking prospect is the $(m + 1)th$ person to be interviewed. What is the probability that the highest rank candidate among the first $m$ candidates is one of the first $k$ candidates who were rejected?
[b](c)[/b] (6 points:) What is the probability that the prospect with the highest rank is the $(m+1)th$ person and that Juli will choose the $(m+1)th$ man using this algorithm?
[b](d)[/b] (16 points:) The total probability that Juli will choose the highest ranking prospect among the $1000$ prospects is the sum of the probability for each possible value of $m+1$ with $m+1$ ranging between $k+1$ and $1000$.
Find the sum. To simplify your answer use the formula
$In N \approx \frac{1}{N-1}+\frac{1}{N-2}+...+\frac{1}{2}+1$
[b](e)[/b] (6 points:) Find that value of $k$ that maximizes the probability of choosing the highest ranking prospect without interviewing all $1000$ candidates. You may need to know that the maximum of the function $x ln \frac{A}{x-1}$ is approximately $\frac{A + 1}{e}$, where $A$ is a constant and $e$ is Euler’s number, $e = 2.718....$
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
In the tetrahedron $OABC$ we denote by $\alpha,\beta,\gamma$ the measures of the angles $\angle BOC,\angle COA,$ and $\angle AOB$, respectively. Prove the inequality
\[\cos^2\alpha+\cos^2\beta+\cos^2\gamma<1+2\cos\alpha\cos\beta\cos\gamma \]
A [i]unitary [/i] divisor d of a number $n$ is a divisor $n$ that has the property $\gcd (d, n/d) = 1$. If $n = 1620$, what is the sum of all of the unitary divisors of $d$?
Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A,$ points $B'$ and $C'$ lie on the rays $AB$ and $AC,$ respectively , so that points $B',$ $C'$ and $D$ are collinear and points $B,$ $C,$ $B'$ and $C'$ lie on one circle with center $O.$ Prove that if $M$ is the midpoint of $BC$ and $H$ is the orthocenter of $ABC,$ then $DHMO$ is a parallelogram.
A triangle with side lengths $2$ and $3$ has an area of $3$. Compute the third side length of the triangle.
Solve the equation $x^2+ 2 = 4\sqrt{x^3+1}$
In a country there are $n>100$ cities and initially no roads. The government randomly determined the cost of building a two-way road between any two cities, using all amounts from $1$ to $\frac{n(n-1)}{2}$ thalers once (all options are equally likely). The mayor of each city chooses the cheapest of the $n-1$ roads emanating from that city and it is built (this may be the mutual desired of the mayors of both cities being connected, or only one of the two).
After the construction of these roads, the cities are divided into $M$ connected components (between cities of the same connected component, you can get along the constructed roads, possibly via other cities, but this is not possible for cities of different components). Find the expected value of the random variable $M$.
[i]Proposed by F. Petrov[/i]
Let $ABC$ be a scalene triangle with circumcentre $O$, incentre $I$ and orthocentre $H$. Let the second intersection point of circle which passes through $O$ and tangent to $IH$ at point $I$, and the circle which passes through $H$ and tangent to $IO$ at point $I$ be $M$. Prove that $M$ lies on circumcircle of $ABC$.
In $4$-dimensional space, a set of $1 \times 2 \times 3 \times 4$ bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks:
[list]i) $2 \times 5 \times 7 \times 12$
ii) $5 \times 5 \times 10 \times 12$
iii) $6 \times 6 \times 6 \times 6$.[/list]
Consider these two geoboard quadrilaterals. Which of the following statements is true?
[asy]
for (int a = 0; a < 5; ++a)
{
for (int b = 0; b < 5; ++b)
{
dot((a,b));
}
}
draw((0,3)--(0,4)--(1,3)--(1,2)--cycle);
draw((2,1)--(4,2)--(3,1)--(3,0)--cycle);
label("I",(0.4,3),E);
label("II",(2.9,1),W);
[/asy]
$\text{(A)}\ \text{The area of quadrilateral I is more than the area of quadrilateral II.}$
$\text{(B)}\ \text{The area of quadrilateral I is less than the area of quadrilateral II.}$
$\text{(C)}\ \text{The quadrilaterals have the same area and the same perimeter.}$
$\text{(D)}\ \text{The quadrilaterals have the same area, but the perimeter of I is more than the perimeter of II.}$
$\text{(E)}\ \text{The quadrilaterals have the same area, but the perimeter of I is less than the perimeter of II.}$
Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.
In a unit cube $ABCD - EFGH$, an equilateral triangle $BDG$ cuts out a circle from the circumsphere of the cube. Find the area of the circle.
One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of this triangle to the area of the quadrilateral formed by three such lines and one of the sides of the triangle.
Is it possible to color the cells of a table 19 x 19 in yellow, blue, red, and green so that each rectangle $a$ x $b$ ($a,b\geq 2$) in the table has at least 2 cells in different color?