Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that \[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]

Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.

draNx rolls 1412 fair 6-sided dice. What is the probability the sum is in the range [4942, 5000]? Your score is determined by the function $max\{0, 20 - 200|A - E|\}$ where $A$ is the actual answer, and $E$ is your estimate. [i]Lightning 5.2[/i]

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

When Ann meets new people, she tries to find out who is acquainted with who. In order to memorize it she draws a circle in which each person is depicted by a chord; moreover, chords corresponding to acquainted persons intersect (possibly at the ends), while the chords corresponding to non-acquainted persons do not. Ann believes that such set of chords exists for any company. Is her judgement correct? (5)

In triangle $ABC$ the bisector of the angle at $B$ meets $AC$ at $D$ and the bisector of the angle at $C$ meets $AB$ at $E$. These bisectors intersect at $O$ and the lengths of $OD$ and $OE$ are equal. Prove that either $\angle BAC = 60^o$ or triangle $ABC$ is isosceles.

If $\frac{a}{3}=b$ and $\frac{b}{4}=c$, what is the value of $\frac{ab}{c^2}$? $\text{(A) }12\qquad\text{(B) }36\qquad\text{(C) }48\qquad\text{(D) }60\qquad\text{(E) }144$

In an isosceles triangle $ABC$ with an angle $\angle A= 20^o$ and base $BC=12$ point $E$ on the side $AC$ is chosen such that $\angle ABE= 30^o$ , and point $F$ on the side $AB$ such that $EF = FC$ . Find the length of $FC$.

Let $c_0,\,c_1,\,c_2,\,\ldots$ be a sequence defined so that \[ \frac{1-3x-\sqrt{1-14x+9x^2}}{4}=\sum_{k=0}^\infty c_kx^k \] for sufficiently small $x$. For a positive integer $n$, let $A$ be the $n$-by-$n$ matrix with $i,j$-entry $c_{i+j-1}$ for $i$ and $j$ in $\{1,\,\ldots,\,n\}$. Find the determinant of $A$.

Find the side lengths of the triangle $ABC$ with area $S$ and $\angle BAC = x$ such that the side $BC$ is as short as possible.

Let $ A \equal{} (a_{ij})$, where $ i,j \equal{} 1,2,\ldots,n$, be a square matrix with all $ a_{ij}$ non-negative integers. For each $ i,j$ such that $ a_{ij} \equal{} 0$, the sum of the elements in the $ i$th row and the $ j$th column is at least $ n$. Prove that the sum of all the elements in the matrix is at least $ \frac {n^2}{2}$.

Let $p>2$ be a prime number and $n$ a positive integer. Prove that $pn^2$ has at most one positive divisor $d$ for which $n^2+d$ is a square number.

Determine the greatest possible value of n that satisfies the following condition: for any choice of $n$ subsets $M_1, ...,M_n$ of the set $M = \{1,2,...,n\}$ satisfying the conditions i) $i \in M_i$ and ii) $i \in M_j \Leftrightarrow j \notin M_i$ for all $i \ne j$, there exist $M_k$ and $M_l$ such that $M_k \cup M_l = M$. (Moscow regional olympiad,adopted)

[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum. [b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of the triangle, as depicted below. Find $s$. [img]https://cdn.artofproblemsolving.com/attachments/4/3/3b68d42e96717c83bd7fa64a2c3b0bf47301d4.png[/img] [b]p3.[/b] Color a $4\times 7$ rectangle so that each of its $28$ unit squares is either red or green. Show that no matter how this is done, there will be two columns and two rows, so that the four squares occurring at the intersection of a selected row with a selected column all have the same color. [b]p4.[/b] (a) Show that the $y$-intercept of the line through any two distinct points of the graph of $f(x) = x^2$ is $-1$ times the product of the $x$-coordinates of the two points. (b) Find all real valued functions with the property that the $y$-intercept of the line through any two distinct points of its graph is $-1$ times the product of the $x$-coordinates. Prove that you have found all such functions and that all functions you have found have this property. [b]p5.[/b] Let $n$ be a positive integer. We consider sets $A \subseteq \{1, 2,..., n\}$ with the property that the equation $x+y=z$ has no solution with $x\in A$, $y \in A$, $z \in A$. (a) Show that there is a set $A$ as described above that contains $[(n + l)/2]$ members where $[x]$ denotes the largest integer less than or equal to $x$. (b) Show that if $A$ has the property described above, then the number of members of $A$ is less than or equal to $[(n + l)/2]$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The incircle $(I)$ of a triangle $ABC$ touches $BC,CA,AB$ at $D,E, F$. Let $ID, IE, IF$ intersect $EF, FD,DE$ at $X,Y,Z$, respectively. The lines $\ell_a, \ell_b, \ell_c$ through $A,B,C$ respectively and are perpendicular to $YZ,ZX,XY$ . Prove that $\ell_a, \ell_b, \ell_c$ are concurrent at a point that is on the line segment joining $I$ and the centroid of triangle $ABC$ . Nguyễn Minh Hà

Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$

In a sequence of natural numbers, the first number is $a$, and each subsequent number is the smallest number coprime to all the previous ones and greater than all of them. Prove that in this sequence from some place all numbers will be primes.

Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$

A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0. a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared". b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 1) $e-2$ , $e$ , $e+4$ are all prime numbers. 2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$ Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$ Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$ For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$? $\textbf{(A) }\$179.95\qquad \textbf{(B) }\$199.95\qquad \textbf{(C) }\$219.95\qquad \textbf{(D) }\$239.95\qquad \textbf{(E) }\$259.95\qquad$

Given a subset of $\{1,2,...,n\}$, we define its [i]alternating sum [/i] in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the [i]alternating sum[/i] of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$. Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.

Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds: For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that \[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\] contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).

Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

$S={1,2,...,1000}$ and $T'=\left\{ 1001-t|t \in T\right\}$. A set $P$ satisfies the following three conditions: $1.$ All elements of $P$ are a subset of $S$. $2. A,B \in P \Rightarrow A \cap B \neq \O$ $3. A \in P \Rightarrow A' \in P$ Find the maximum of $|P|$.