Let $ M,N,P,Q $ be the middlepoints of the segments $ AB,BC,CD,DA, $ respectively, of a convex quadrilateral $ ABCD. $ Prove that if $ ANP $ and $ CMQ $ are equilateral, then $ ABDC $ is a rhombus . Moreover, determine the angles of this rhombus.

$$\frac{p}{q} = \sum_{n = 1}^\infty \frac{1}{2^{n + 6}} \frac{(10 - 4\cos^2(\frac{\pi n}{24})) (1 - (-1)^n) - 3\cos(\frac{\pi n}{24}) (1 + (-1)^n)}{25 - 16\cos^2(\frac{\pi n}{24})}$$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown on the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for $k \in \left \{1,2,3,4,5 \right \}$. The covering satisfies the following conditions: $1)$ It is possible not to use figures of some type and it is possible to use several figures of the same type. The unit triangles in the figures correspond to the unit triangles in the grid. $2)$ Every unit triangle in the grid is covered, no two figures overlap and every figure is fully contained in $T$. Determine the smallest possible number of figures of type $1$ that can be used in such a covering. [i]Proposed by Ilija Jovcheski[/i]

In triangle $ABC$, angle $A$ is equal to $a$ and angle $B$ is equal to $2a$. A circle with center at point $C$ of radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$, at points $M$ and $N$. Find the angles of triangle $MAN$.

Let $ABXC$ be a parallelogram. Points $K,P,Q$ lie on $\overline{BC}$ in this order such that $BK = \frac{1}{3} KC$ and $BP = PQ = QC = \frac{1}{3} BC$. Rays $XP$ and $XQ$ meet $\overline{AB}$ and $\overline{AC}$ at $D$ and $E$, respectively. Suppose that $\overline{AK} \perp \overline{BC}$, $EK-DK=9$ and $BC=60$. Find $AB+AC$. [i]Proposed by Evan Chen[/i]

Suppose that one of every $500$ people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate; in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from the population and gets a positive test result actually has the disease. Which of the following is closest to $p$? $ \textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$

When the mean, median, and mode of the list $10,2,5,2,4,2,x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? $\text{(A)}\ 3\qquad\text{(B)}\ 6 \qquad\text{(C)}\ 9 \qquad\text{(D)}\ 17\qquad\text{(E)}\ 20$

The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.

Find the sum of all positive integers $n$ such that $1+2+\cdots+n$ divides \[15\left[(n+1)^2+(n+2)^2+\cdots+(2n)^2\right].\]

[b]p1.[/b] A rectangle $ABCD$ is cut from a piece of paper and folded along a straight line so that the diagonally opposite vertices $A$ and $C$ coincide. Find the length of the resulting crease in terms of the length ($\ell$) and width ($w$) of the rectangle. (Justify your answer.) [b]p2.[/b] The residents of Andromeda use only bills of denominations $\$3 $and $\$5$ . All payments are made exactly, with no change given. What whole-dollar payments are not possible? (Justify your answer.) [b]p3.[/b] A set consists of $21$ objects with (positive) weights $w_1, w_2, w_3, ..., w_{21}$ . Whenever any subset of $10$ objects is selected, then there is a subset consisting of either $10$ or $11$ of the remaining objects such that the two subsets have equal fotal weights. Find all possible weights for the objects. (Justify your answer.) [b]p4.[/b] Let $P(x) = x^3 + x^2 - 1$ and $Q(x) = x^3 - x - 1$ . Given that $r$ and $s$ are two distinct solutions of $P(x) = 0$ , prove that $rs$ is a solution of $Q(x) = 0$ [b]p5.[/b] Given: $\vartriangle ABC$ with points $A_1$ and $A_2$ on $BC$ , $B_1$ and $B_2$ on $CA$, and $C_1$ and $C_2$ on $AB$. $A_1 , A_2, B_1 , B_2$ are on a circle, $B_1 , B_2, C_1 , C_2$ are on a circle, and $C_1 , C_2, A_1 , A_2$ are on a circle. The centers of these circles lie in the interior of the triangle. Prove: All six points $A_1$ , $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are on a circle. [img]https://cdn.artofproblemsolving.com/attachments/7/2/2b99ddf4f258232c910c062e4190d8617af6fa.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for positive reals $a,b,c$, \[\frac{8a^2+2ab}{(b+\sqrt{6ac}+3c)^2}+\frac{2b^2+3bc}{(3c+\sqrt{2ab}+2a)^2}+\frac{18c^2+6ac}{(2a+\sqrt{3bc}+b})^2\geq 1\]

We build rhombuses from natural numbers. Find the sum of the numbers in the $n$-th rhombus. [img]https://cdn.artofproblemsolving.com/attachments/e/7/22360573f76c615ca43bbacb8f15e587772ca4.png[/img]

Let two circles $C_1,C_2$ with different radius be externally tangent at a point $T$. Let $A$ be on $C_1$ and $B$ be on $C_2$, with $A,B \ne T$ such that $\angle ATB = 90^o$. (a) Prove that all such lines $AB$ are concurrent. (b) Find the locus of the midpoints of all such segments $AB$.

Sum of $ 100 $ natural distinct numbers is $ 9999 $. Prove that $ 2006 $ divide their product.

If $f$ is a continuous real function such that $ f(x-1) + f(x+1) \ge x + f(x) $ for all $x$, what is the minimum possible value of $ \displaystyle\int_{1}^{2005} f(x) \, \mathrm{d}x $?

Let $x$ and $y$ be positive real numbers such that: $x+y^{2016}\geq 1$. Prove that $x^{2016}+y> 1-\frac{1}{100}$

Cyrus the frog jumps 2 units in a direction, then 2 more in another direction. What is the probability that he lands less than 1 unit away from his starting position? (I forgot answer choices)

[asy] real t=pi/12;real u=8*t; real cu=cos(u);real su=sin(u); draw(unitcircle); draw((cos(-t),sin(-t))--(cos(13*t),sin(13*t))); draw((cu,su)--(cu,-su)); label("A",(cos(13*t),sin(13*t)),W); label("B",(cos(-t),sin(-t)),E); label("C",(cu,su),N); label("D",(cu,-su),S); label("E",(cu,sin(-t)),NE); label("2",((cu-1)/2,sin(-t)),N); label("6",((cu+1)/2,sin(-t)),N); label("3",(cu,(sin(-t)-su)/2),E); //Credit to Zimbalono for the diagram[/asy] Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is $\textbf{(A) }4\sqrt{5}\qquad\textbf{(B) }\sqrt{65}\qquad\textbf{(C) }2\sqrt{17}\qquad\textbf{(D) }3\sqrt{7}\qquad \textbf{(E) }6\sqrt{2}$

For how many integers $n$ does the expression \[\sqrt{\frac{\log (n^2) - (\log n)^2}{\log n - 3}} \] represent a real number, where log denotes the base $10$ logarithm? $ \textbf{(A) }900 \qquad \textbf{(B) }2\qquad \textbf{(C) }902 \qquad \textbf{(D) } 2 \qquad \textbf{(E) }901$

Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$.

An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?

The sequence $a_1,a_2,\ldots$ is defined by $a_1=2019$, $a_2=2020$, $a_3=2021$, $a_{n+3}=a_n(a_{n+1}a_{n+2}+1)$ for $n\ge 1$. Determine the value of the infinite sum \[\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots \]

Given a segment $AB$ of the length 1, define the set $M$ of points in the following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$