$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly. Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.

A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is [i]UMD certified[/i] if each of these $16$ squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its center. How many distinct UMD certified quilts are there? \[\rm a. ~33\qquad \mathrm b. ~36 \qquad \mathrm c. ~45\qquad\mathrm d. ~54\qquad\mathrm e. ~81\]

Five points are chosen uniformly and independently at random on the surface of a sphere. Next, 2 of these 5 points are randomly picked, with every pair equally likely. What is the probability that the 2 points are separated by the plane containing the other 3 points?

Let $S$ be the set of all subsets of the positive integers. Construct a function $f \colon \mathbb{R} \rightarrow S$ such that $f(a)$ is a proper subset of $f(b)$ whenever $a <b.$

For which integers $n \geq 2$ is it possible to draw $n$ straight lines in the plane in such a way that there are at least $n - 2$ points where exactly three of the lines meet?

Let $ABC$ be a triangle inscribed in circle $(O)$ with diamter $KL$ passes through the midpoint $M$ of $AB$ such that $L, C$ lie on the different sides respect to $AB$. A circle passes through $M, K $cuts $LC$ at$ P, Q $(point $P$ lies between$ Q, C$). The line $KQ $cuts $(LMQ)$ at $R$. Prove that $ARBP$ is cyclic and$ AB$ is the symmedian of triangle $APR$. Please help :)

Talya went for a 6 kilometer run. She ran 2 kilometers at 12 kilometers per hour followed by 2 kilometers at 10 kilometers per hour followed by 2 kilometers at 8 kilometers per hour. Talya’s average speed for the 6 kilometer run was $\frac{m}{n}$ kilometers per hour, where m and n are relatively prime positive integers. Find m + n.

Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.

Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. Prove that if $AP=4PN$, then $ABCD$ is a parallelogram.

A non-empty subset of $\{1,2, ..., n\}$ is called [i]arabic [/i] if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of $\{1,2, ..., n\}$ has the same parity as $n$.

A circle of radius $10$ is tangent to two adjacent sides of a square and intersects its two remaining sides at the endpoints of a diameter of the circle. Find the side length of the square.

The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

A palindrome between $ 1000$ and $ 10,000$ is chosen at random. What is the probability that it is divisible by $ 7?$ $ \textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6}\qquad \textbf{(E)}\ \dfrac{1}{5}$

On the diagonal $AC$ of the convex quadrilateral $ABCD$ points $P$,$Q$ are chosen such that triangles $ABD$,$PCD$ and $QBD$ are similar to each other in this order. Prove that $AQ=PC$ [i]M. Zorka[/i]

The sphere $ \omega $ passes through the vertex $S$ of the pyramid $SABC$ and intersects with the edges $SA,SB,SC$ at $A_1,B_1,C_1$ other than $S$. The sphere $ \Omega $ is the circumsphere of the pyramid $SABC$ and intersects with $ \omega $ circumferential, lies on a plane which parallel to the plane $(ABC)$. Points $A_2,B_2,C_2$ are symmetry points of the points $A_1,B_1,C_1$ respect to midpoints of the edges $SA,SB,SC$ respectively. Prove that the points $A$, $B$, $C$, $A_2$, $B_2$, and $C_2$ lie on a sphere.

In triangle $ABC$, the incenter is $I$ and the circumcenter is $O$. Let $AI$ intersects $(ABC)$ second time at $P$ . The line passes through $I$ and perpendicular to $AI$ intersects $BC$ at $X$. The feet of the perpendicular from $X$ to $IO$ is $Y$. Prove that $A,P,X,Y$ cyclic.

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

Consider an acutangle triangle $ABC$ with circumcenter $O$. A circumference that passes through $B$ and $O$ intersect sides $BC$ and $AB$ in points $P$ and $Q$. Prove that the orthocenter of triangle $OPQ$ is on $AC$.

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

If $z\in\mathbb{C}$ is a solution of the equation $$x^n+a_1x^{n-1}+a_2x^{n-2}+\ldots+a_n=0$$ with real coefficients $0<a_n\leq a_{n-1}\leq\ldots\leq a_1<1$, show that $|z|<1$.

In the grid below, draw horizontal and vertical segments of unit length joining pairs of adjacent dots (some have been given to you) so that $1.$ every dot is connected by line segments to exactly $1$ or $3$ adjacent dots, $2.$ any dot can be reached from any other dot by following a path of segments, and $3.$ no area is completely enclosed by segments. Note: “Unit length” is the length between two adjacent dots when there is no missing dot between them. For example, we cannot draw a vertical line segment down from the dot in the top right corner because the length of this segment would be 2 units. There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] unitsize(0.5cm); for (int i = 0; i <= 6; ++i) { for (int j = 0; j <= 6; ++j) { if ((i == 2 && j == 0) != true && (i == 6 && j == 0) != true && (i == 6 && j == 3) != true && (i == 0 && j == 5) != true && (i == 6 && j == 5) != true) { dot((i, j)); } } } draw((0, 3) -- (0,4)); draw((0, 3) -- (1,3)); draw((1,3) -- (1,4)); draw((2,2) -- (3,2)); draw((3,2) -- (3,3)); draw((3,3) -- (2,3)); draw((2,3) -- (2,4)); draw((2,4) -- (3,4)); draw((4,2) -- (4,1)); draw((4,1) -- (5,1)); draw((5,1) -- (5,2)); draw((5,2) -- (5,3)); draw((5,3) -- (4,3)); [/asy]

Let $ r$ be a positive rational. Prove that $\frac{8r + 21}{3r + 8}$ is a better approximation to $\sqrt7$ that $ r$.

Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is $80$ cm tall, $120$ cm wide and its corners are denoted by $A, B, C$ and $D$ in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and $B$. The third vertex of the yellow foil is $S$, the trisecting point of side $DC$ closer to $D$, whereas the third vertex of the red foil is $P$, which is one fourth on the way on segment $SC$, closer to $C$. The red region (i.e. triangle $BPE$) is of area $16$ dm$^2$. What is the total area of the regions not covered by foil? [img]https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png[/img]

Find all pairs $(x,y)$ of real numbers that satisfy the system \begin{align*} x \cdot \sqrt{1-y^2} &=\frac14 \left(\sqrt3+1 \right), \\ y \cdot \sqrt{1-x^2} &= \frac14 \left( \sqrt3 -1 \right). \end{align*}

Suppose there is an infinite sequence of lights numbered $1, 2, 3,...,$ and you know the following two rules about how the lights work: $\bullet$ If the light numbered $k$ is on, the lights numbered $2k$ and $2k + 1$ are also guaranteed to be on. $\bullet$ If the light numbered $k$ is off, then the lights numbered $4k + 1$ and $4k + 3$ are also guaranteed to be off. Suppose you notice that light number $2023$ is on. Identify all the lights that are guaranteed to be on?