The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$. (Karl Czakler)

A triangle with vertices $(6,5)$, $(8,-3)$, and $(9,1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles? $\textbf{(A) }9\qquad \textbf{(B) }\dfrac{28}{3}\qquad \textbf{(C) }10\qquad \textbf{(D) }\dfrac{31}{3}\qquad \textbf{(E) }\dfrac{32}{3}\qquad$

Given an $ABC$ acute triangle with $O$ the center of the circumscribed circle. Suppose that $\omega$ is a circle that is tangent to the line $AO$ at point $A$ and also tangent to the line $BC$. Prove that $\omega$ is also tangent to the circumcircle of the triangle $BOC$.

Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL \equal{} \angle XDC \equal{} 30^\circ$ and $ CX \equal{} O_1O_2$. [i] Author: Arnoldo Aguilar (El Salvador)[/i]

Compute the radius of the largest circle that fits entirely within a unit cube.

Let the point $P$ in the interior of the triangle $ABC$. $(AP, (BP, (CP$ intersect the circumcircle of $ABC$ at $A_{1}, B_{1}, C_{1}$. Prove that the maximal value of the sum of the areas $A_{1}BC$, $B_{1}AC$, $C_{1}AB$ is $p(R-r)$, where $p, r, R$ are the usual notations for the triangle $ABC$.

Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$ b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$ (D. Pirshtuk)

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

[b]p1.[/b] It can be shown in Calculus that the area between the x-axis and the parabola $y=kx^2$ (к is a positive constant) on the $x$-interval $0 \le x \le a$ is $\frac{ka^3}{3}$ a) Find the area between the parabola $y=4x^2$ and the x-axis for $0 \le x \le 3$. b) Find the area between the parabola $y=5x^2$ and the x-axis for $-2 \le x \le 4$. c) A square $2$ by $2$ dartboard is situated in the $xy$-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point $(0, 1)$ than to the bottom edge of the dartboard. [b]p2.[/b] When two rows of a determinant are interchanged, the value of the determinant changes sign. There are also certain operations which can be performed on a determinant which leave its value unchanged. Two such operations are changing any row by adding a constant multiple of another row to it, and changing any column by adding a constant multiple of another column to it. Often these operations are used to generate lots of zeroes in a determinant in order to simplify computations. In fact, if we can generate zeroes everywhere below the main diagonal in a determinant, the value of the determinant is just the product of all the entries on that main diagonal. For example, given the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 6 & 2 \\ 3 & 10 & 4 \end{vmatrix}$ we add $-2$ times the first row to the second row, then add $-2$ times the second row to the third row, giving the new determinant $\begin{vmatrix} 1 & 2 & 3 \\ 0 & 2 & -4 \\ 0 & 0 & 3 \end{vmatrix}$ , and the value is the product of the diagonal entries: $6$. a) Transform this determinant into another determinant with zeroes everywhere below the main diagonal, and find its value: $\begin{vmatrix} 1 & 3 & -1 \\ 4 & 7 & 2 \\ 3 & -6 & 5 \end{vmatrix}$ b) Do the same for this determinant: $\begin{vmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{vmatrix}$ [b]p3.[/b] In Pascal’s triangle, the entries at the ends of each row are both $1$, and otherwise each entry is the sum of the two entries diagonally above it: Row Number $0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1$ $1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\,\,1$ $2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\, 2 \,\,1$ $3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1\,\, 3 \,\, 3 \,\, 1$ $4\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \,\,4 \,\, 6 \,\, 4 \,\, 1$ $...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...$ This triangle gives the binomial coefficients in expansions like $( a + b)^3 = 1a^3 + 3a^2 b + 3 ab^2 + 1b^3$ . a) What is the sum of the numbers in row #$5$ of Pascal's triangle? b) What is the sum of the numbers in row #$n$ of Pascal's triangle? c) Show that in row #$6$ of Pascal's triangle, the sum of all the numbers is exactly twice the sum of the first, third, fifth, and seventh numbers in the row. d) Prove that in row #$n$ of Pascal's triangle, the sum of ail the numbers is exactly twice the sum of the numbers in the odd positions of that row. [b]p4.[/b] The product: of several terms is sometimes described using the symbol $\Pi$ which is capital pi, the Greek equivalent of $p$, for the word "product". For example the symbol $\prod^4_{k=1}(2k +1)$ means the product of numbers of the form $(2k + 1)$, for $k=1,2,3,4$. Thus it equals $945$. a) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k}{k + 2}$ b) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k^2 + 10k+ 17}{k^2+4k + 41}$ c) Evaluate as a reduced fraction $\prod_{k=1}^{\infty}\frac{k^3-1}{k^3+1}$ [b]p5.[/b] a) In right triangle $CAB$, the median $AF$, the angle bisector $AE$, and the altitude $AD$ divide the right angld $A$ into four equal angles. If $AB = 1$, find the area of triangle $AFE$. [img]https://cdn.artofproblemsolving.com/attachments/5/1/0d4a83e58a65c2546ce25d1081b99d45e30729.png[/img] b) If in any triangle, an angle is divided into four equal angles by the median, angle bisector, and altitude drawn from that angle, prove that the angle must be a right angle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$, and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Consider the circumference $\Gamma$ with with diameter $BC$. A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$, cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$), and cuts the side $AB$ at $N$. Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\Omega$ and let diagonals $AC$ and $BD$ intersect at $X$. Suppose that $AEFB$ is inscribed in a circumcircle of triangle $ABX$ such that $EF$ and $AB$ are parallel. $FX$ meets the circumcircle of triangle $CDX$ again at $G$. Let $EX$ meets $AB$ at $P$, and $XG$ meets $CD$ at $Q$. Denote by $S$ the intersection of the perpendicular bisector of $\overline{EG}$ and $\Omega$ such that $S$ is closer to $A$ than $B$. Prove that line through $S$ parallel to $PQ$ is tangent to $\Omega$.

Source: 1976 Euclid Part A Problem 9 ----- A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. The measure of the angle subtended at the centre of the circle by the shortest side is $\textbf{(A) } 30 \qquad \textbf{(B) } 45 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } \text{none of these}$

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.

Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.

On a cube, 27 points are marked in the following manner: one point in each corner, one point on the middle of each edge, one point on the middle of each face, and one in the middle the cube. The number of lines containing three out of these points is A. 33 B. 42 C. 49 D. 72 E. 81

[b]8.1 [/b] Construct a quadrilateral using side lengths and distances between the midpoints of the diagonals. [b]8.2[/b] It is known that $a,b$ and $\sqrt{a}+\sqrt{b} $ are rational numbers. Prove that then $\sqrt{a}$, $\sqrt{b} $ are rational. [b]8.3 / 9.2[/b] Solve equation $x^3 - [x]=3$ [b]8.4[/b] Prove that if in a triangle the angle bisector of the vertex, bisects the angle between the median and the altitude, then the triangle either isosceles or right. . [b]8.5[/b] Given $n$ numbers $x_1, x_2, . . . , x_n$, each of which is equal to $+1$ or $-1$. At the same time $$x_1x_2 + x_2x_3 + . . . + x_{n-1}x_n + x_nx_1 = 0 .$$ Prove that $n$ is divisible by $4$. [b]8.6[/b] There are $n$ points marked on the circle, and it is known that for of any two, one of the arcs connecting them has a measure less than $120^0$.Prove that all points lie on an arc of size $120^0$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983442_1961_leningrad_math_olympiad]here[/url].

Determine whether or not the length of symmedian is not greater than the length of the angle bisector drawn from the same vertex?

Let $ABC$ be a triangle and let $\Omega$ denote the circumcircle of $ABC$. The foot of altitude from $A$ to $BC$ is $D$. The foot of altitudes from $D$ to $AB$ and $AC$ are $K;L$ , respectively. Let $KL$ intersect $\Omega$ at $X;Y$, and let $AD$ intersect $\Omega$ at $Z$. Prove that $D$ is the incenter of triangle $XYZ$

Let $a_1,b_1,c_1$ are three lines each two of them are mutually crossed and aren't parallel to some plane. The lines $a_2,b_2,c_2$ intersect the lines $a_1,b_1,c_1$ at the points $a_2$ in $A$, $C_2$, $B_1$; $b_2$ in $C_1$, $B$, $A_2$; $c_2$ in $B_2$, $A_1$, $C$ respectively in such a way that $A$ is the perpendicular bisector of $B_1C_2$, $B$ is the perpendicular bisector of $C_1A_2$ and $C$ is the perpendicular bisector of $A_1B_2$. Prove that: (a) $A$ is the perpendicular bisector of $B_2C_1$, $B$ is the perpendicular bisector of $C_2A_1$ and $C$ is the perpendicular bisector of $A_2B_1$; (b) triangles $A_1B_1C_1$ and $A_2B_2C_2$ are the same.

Prove that, for any integers $a,b,c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.

We are given a triangle $ABC$ and three rectangles $R_1,R_2,R_3$ with sides parallel to two fixed perpendicular directions and such that their union covers the sides $AB,BC$, and $CA$; i.e., each point on the perimeter of $ABC$ is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the union of $R_1,R_2,R_3.$

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? $ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$