In triangle $ABC$ with $AB=23$, $AC=27$, and $BC=20$, let $D$ be the foot of the $A$ altitude. Let $\mathcal{P}$ be the parabola with focus $A$ passing through $B$ and $C$, and denote by $T$ the intersection point of $AD$ with the directrix of $\mathcal P$. Determine the value of $DT^2-DA^2$. (Recall that a parabola $\mathcal P$ is the set of points which are equidistant from a point, called the $\textit{focus}$ of $\mathcal P$, and a line, called the $\textit{directrix}$ of $\mathcal P$.)

For how many integers $n$, there are four distinct real numbers satisfying the equation $ |x^2-4x-7|=n$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 5 $

A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.

One afternoon a bakery finds that it has $300$ cups of flour and $300$ cups of sugar on hand. Annie and Sam decide to use this to make and sell some batches of cookies and some cakes. Each batch of cookies will require $1$ cup of flour and $3$ cups of sugar. Each cake will require $2$ cups of flour and $1$ cup of sugar. Annie thinks that each batch of cookies should sell for $2$ dollars and each cake for $1$ dollar, but Sam thinks that each batch of cookies should sell for $1$ dollar and each cake should sell for $3$ dollars. Find the difference between the maximum dollars of income they can receive if they use Sam's selling plan and the maximum dollars of income they can receive if they use Annie's selling plan.

Let $x,y,z\geq0$ be real numbers such that $x+y+z=1$ Define $f(x,y,z)$ in this way : \[f(x,y,z)=\frac{x(2y-z)}{1+x+3y}+\frac{y(2z-x)}{1+y+3z}+\frac{z(2x-y)}{1+z+3x}\] Find the minimum value and maximum value of $f(x,y,z)$ .

Three circles of radius $1$ are mutually tangent, as shown. What is the area of the triangle whose vertices are the points of tangency?

Let $BE$ and $CF$ be the altitudes of acute triangle $ABC$. Let $H$ be the orthocenter of $ABC$ and $M$ be the midpoint of side $BC$. The points of intersection of the midperpendicular line to $BC$ with segments $BE$ and $CF$ are denoted by $K$ and $L$ respectively. The point $Q$ is the orthocenter of triangle $KLH$. Prove that $Q$ belongs to the median $AM$. (Bohdan Zheliabovskyi)

$1992$ vectors are given in the plane. Two players pick unpicked vectors alternately. The winner is the one whose vectors sum to a vector with larger magnitude (or they draw if the magnitudes are the same). Can the first player always avoid losing?

A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$. $\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$. $\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.

An integer consists of 7 different digits, and is a multiple of each of its digits. What digits are in this nubmer?

For a real number $x$, the minimum value of the expression $$\frac{2x^2 + x - 3}{x^2 - 2x + 3}$$ can be written in the form $\frac{a-\sqrt{b}}{c}$, where $a, b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$

Let $d$ be a positive integer and let $A$ be a $d\times d$ matrix with integer entries. Suppose $I+A+A_2+\ldots+A_{100}=0$ (where $I$ denotes the identity $d\times d$ matrix, and $0$ denotes the zero matrix, which has all entries $0$). Determine the positive integers $n\le100$ for which $A_n+A_{n+1}+\ldots+A_{100}$ has determinant $\pm1$.

Two distinct similar rhombi share a diagonal. The smaller rhombus has area $1$, and the larger rhombus has area $9$. Compute the side length of the larger rhombus.

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

Prove that the determinant of the matrix $$\begin{pmatrix} a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\ a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k \end{pmatrix}$$ is divisible by $k^{n-1}$ and find its other factor.

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a,b,c$ such that $a+b,a+c,b+c$ are perfect squares.

[b]8.1[/b] In the parallelogram $ABCD$ , the diagonal $AC$ is greater than the diagonal $BD$. The point $M$ on the diagonal $AC$ is such that around the quadrilateral $BCDM$ one can circumscribe a circle. Prove that $BD$ is the common tangent of the circles circumscribed around the triangles $ABM$ and $ADM$. [img]https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png[/img] [b]8.2 [/b] $A$ is an odd integer, $x$ and $y$ are roots of equation $t^2+At-1=0$. Prove that $x^4 + y^4$ and $x^5+ y^5$ are coprime integer numbers. [b]8.3[/b] A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made. [b]8.4 /7.6[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. [b]8.5 [/b] All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way. [url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6*[/url] (asterisk problems in separate posts) PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Aisling and Brendan take alternate moves in the following game. Before the game starts, the number $x = 2023$ is written on a piece of paper. Aisling makes the first move. A move from a positive integer $x$ consists of replacing $x$ either with $x + 1$ or with $x/p$ where $p$ is a prime factor of $x$. The winner is the first player to write $x = 1$. Determine whether Aisling or Brendan has a winning strategy for this game.

Let be four functions $ f,g,s,i:\mathbb{N}\longrightarrow\mathbb{N} $ such that $ s(x)=\max (f(x),g(x)) $ and $ i(x)=\min (f(x),g(x)) , $ for any natural number $ x. $ Prove that $ f=g $ if $ s $ is surjective and $ i $ injective.

In a plane, 2014 lines are distributed in 3 groups. in every group all the lines are parallel between themselves. What is the maximum number of triangles that can be formed, such that every side of such triangle lie on one of the lines?

$16$ children are sitting at a round table. After the break, they sit down again on table. They find that each child is either sitting on its original [lace or in one of the two neighboring places. How many seating arrangements are possible in this way after the break?

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

Which of the following methods of proving a geometric figure a locus is not correct? $ \textbf{(A)}\ \text{Every point of the locus satisfies the conditions and every point not on the locus does not satisfy the conditions.}$ $ \textbf{(B)}\ \text{Every point not satisfying the conditions is not on the locus and every point on the locus does satisfy the conditions.}$ $ \textbf{(C)}\ \text{Every point satisfying the conditions is on the locus and every point on the locus satisfies the conditions.}$ $ \textbf{(D)}\ \text{Every point not on the locus does not satisfy the conditions and every point not satisfying} \\ \text{the conditions is not on the locus.}$ $ \textbf{(E)}\ \text{Every point satisfying the conditions is on the locus and every point not satisfying the conditions is not on the locus.}$

Let $ S$ be the set of $25$ points $(x, y)$ with $0\le x, y \le 4$. A triangle whose three vertices are in $S$ is chosen at random. What is the expected value of the square of its area?