Let $ABC$ be a triangle, not right-angled, with positive integer angle measures (in degrees) and circumcenter $O$. Say that a triangle $ABC$ is [i]good[/i] if the following three conditions hold: (a) There exists a point $P\neq A$ on side $AB$ such that the circumcircle of $\triangle POA$ is tangent to $BO$. (b) There exists a point $Q\neq A$ on side $AC$ such that the circumcircle of $\triangle QOA$ is tangent to $CO$. (c) The perimeter of $\triangle APQ$ is at least $AB+AC$. Determine the number of ordered triples $(\angle A, \angle B,\angle C)$ for which $\triangle ABC$ is good. [i]Proposed by Vincent Huang[/i]

The ratio of $3$ to the positive number $n$ is the same as the ratio of $n$ to $192.$ Find $n.$

Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeros.

Point $P$ is inside a square $ABCD$ such that $\angle APB = 135^\circ, PC=12,$ and $PD=15.$ Compute the area of this square.

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle? [asy] size(5cm); filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray); draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1)); draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1)); label("$4$", (8,2), E); label("$8$", (4,0), S); label("$5$", (3,11/2), NW); draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(.4)); draw((5,7)--(5+21/100,7-28/100)--(5-7/100,7-49/100)--(5-28/100,7-21/100)--cycle,linewidth(.4)); [/asy] $\textbf{(A) } 15\dfrac{1}{8} \qquad \textbf{(B) } 15\dfrac{3}{8} \qquad \textbf{(C) } 15\dfrac{1}{2} \qquad \textbf{(D) } 15\dfrac{5}{8} \qquad \textbf{(E) } 15\dfrac{7}{8}$

Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$. [i]E. Malinnikova[/i]

Let $Q^+$ denote the set of all positive rational number and let $\alpha\in Q^+.$ Determine all functions $f:Q^+ \to (\alpha,+\infty )$ satisfying $$f(\frac{ x+y}{\alpha}) =\frac{ f(x)+f(y)}{\alpha}$$ for all $x,y\in Q^+ .$

Find the sum of all the digits in the decimal representations of all the positive integers less than $1000.$

Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that \[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\] [i]Proposed by Gerhard Wöginger, Austria.[/i]

Given $5$ points in a plane, no three of them being collinear. Each two of these $5$ points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color. [b]a.)[/b] Show that: [i](1)[/i] Among the four segments originating at any of the $5$ points, two are red and two are blue. [i](2)[/i] The red segments form a closed way passing through all $5$ given points. (Similarly for the blue segments.) [b]b.)[/b] Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is [i]nice[/i] if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$. (There can be nonconvex regions as well as regions with more than one boundary curve.)

Angelina drove at an average rate of $ 80$ kph and then stopped $ 20$ minutes for gas. After the stop, she drove at an average rate of $ 100$ kph. Altogether she drove $ 250$ km in a total trip time of $ 3$ hours including the stop. Which equation could be used to solve for the time $ t$ in hours that she drove before her stop? $ \textbf{(A)}\ 80t\plus{}100(8/3\minus{}t)\equal{}250 \qquad \textbf{(B)}\ 80t\equal{}250 \qquad \textbf{(C)}\ 100t\equal{}250 \\ \textbf{(D)}\ 90t\equal{}250 \qquad \textbf{(E)}\ 80(8/3\minus{}t)\plus{}100t\equal{}250$

Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. The point $F$ belongs to the side $AB$, and $DE \perp EF$. The point $G$ lies inside the square, and $GF = FE$ and $GF \perp FE$. Prove that: a) $DE$ is the bisector of the $\angle FDC$ b) $FG$ is the bisector of the $\angle AFD$ c) the point $G$ is the center of the circle inscribed in the triangle $ADF$. (Ercole Suppa, Italy)

Let $ (x_1,x_2,\cdots)$ be a sequence of positive numbers such that $ (8x_2 \minus{} 7x_1)x_1^7 \equal{} 8$ and \[ x_{k \plus{} 1}x_{k \minus{} 1} \minus{} x_k^2 \equal{} \frac {x_{k \minus{} 1}^8 \minus{} x_k^8}{x_k^7x_{k \minus{} 1}^7} \text{ for }k \equal{} 2,3,\ldots \] Determine real number $ a$ such that if $ x_1 > a$, then the sequence is monotonically decreasing, and if $ 0 < x_1 < a$, then the sequence is not monotonic.

Given the circle $c(O,R)$ (with center $O$ and radius $R$), one diameter $AB$ and midpoint $C$ of the arc $AB$. Consider circle $c_1(K,KO)$, where center $K$ lies on the segment $OA$, and consider the tangents $CD,CO$ from the point $C$ to circle $c_1(K,KO)$. Line $KD$ intersects circle $c(O,R)$ at points $E$ and $Z$ (point $E$ lies on the semicircle that lies also point $C$). Lines $EC$ and $CZ$ intersects $AB$ at points $N$ and $M$ respectively. Prove that quadrilateral $EMZN$ is an isosceles trapezoid, inscribed in a circle whose center lie on circle $c(O,R)$.

A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$

Given that $0 < a < b < c < d$, which of the following is the largest? $\textbf{(A)}\ \frac{a\plus{}b}{c\plus{}d} \qquad \textbf{(B)}\ \frac{a\plus{} d}{b\plus{} c} \qquad \textbf{(C)}\ \frac{b\plus{} c}{a\plus{}d}\qquad \textbf{(D)}\ \frac{b\plus{} d}{a\plus{} c} \qquad \textbf{(E)}\ \frac{c\plus{} d}{a\plus{}b}$

For each positive integer \( n \), let \( f(n) \) be the number of ordered triples \( (a, b, c) \) such that \( a, b, c \in \{1, 2, \ldots, n\} \) and that the two roots (possibly equal) of the quadratic equation \( ax^2 + bx + c = 0 \) are both integers. (a) Prove that for every positive real number \( C \), there exists a positive integer \( n_C \) such that for all integers \( n \geq n_C \), we have \( f(n) > C \cdot n \). (b) Prove that for every positive real number \( C \), there exists a positive integer \( n_C \) such that for all integers \( n \geq n_C \), we have \( f(n) < C \cdot n^{\frac{2025}{2024}} \).

Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \cap d_3 = D$, $d_3 \cap d_1 = E,$ $d_1 \cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.

Two acute angles $a$ and $b$ satisfy condition $$\sin^2a+\sin^2b = \sin(a+b)$$ Prove that $a + b = \pi /2$.

Given 12 red points on a circle , find the mininum value of $n$ such that there exists $n$ triangles whose vertex are the red points . Satisfies: every chord whose points are the red points is the edge of one of the $n$ triangles .

In the coordinate system in the plane we consider a convex polygon $W$ and lines given by equations $x = k, y = m$, where $k$ and $m$ are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of $W$ intersects a square if the boundary contains an interior point of the square. Prove that the boundary of $W$ intersects at most $4 \lceil d \rceil $ unit squares, where $d$ is the maximal distance of points belonging to $W$ (i.e., the diameter of $W$) and $\lceil d \rceil$ is the least integer not less than $d.$

Is there a $3 \times 3$ magic square consisting of distinct Fibonacci numbers (both $f_{1}$ and $f_{2}$ may be used; thus two $1$s are allowed)?

Determine all triples of positive integers $a, b, c$ that satisfy a) $[a, b] + [a, c] + [b, c] = [a, b, c]$. b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$. Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.

All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center? $\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$