Found problems: 85335
Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?
Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes.
Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions:
1) There are no two red boxes in the same column or in the same row.
2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column.
[i]Proposed by Christian Reiher[/i]
Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.
A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.
For $n\geq 2$ , an equilateral triangle is divided into $n^2$ congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the $\frac{(n+1)(n+2)}{2}$ vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.
All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$.
[i]Bogdan Enescu[/i]
The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$. Lines $BM$ and $CN$ meet at point $P$. Prove that $PI$ bisects segment $MN$.
Let $n$ be a given even number, $a_1,a_2,\cdots,a_n$ be non-negative real numbers such that $a_1+a_2+\cdots+a_n=1.$ Find the maximum possible value of $\sum_{1\le i<j\le n}\min\{(i-j)^2,(n+i-j)^2\}a_ia_j .$
Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$.
[i]Proposed by Sharvil Kesarwani[/i]
Determine the value of the expression $x^2 + y^2 + z^2$,
if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.
$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?
In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively.
Prove that the lines $DF$, $EG$ and $MN$ are concurrent.
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].
Let $x$, $y$, and $z$ be positive real numbers. Prove that
$\sqrt {\frac {xy}{x^2 + y^2 + 2z^2}} + \sqrt {\frac {yz}{y^2 + z^2 + 2x^2}}+\sqrt {\frac {zx}{z^2 + x^2 + 2y^2}} \le \frac{3}{2}$.
When does equality hold?
Consider a right circular cylinder with radius $ r$ of the base, hight $ h$. Find the volume of the solid by revolving the cylinder about a diameter of the base.
Three squares $BCDE,CAFG$ and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is an isosceles right triangle.
Let $P_{1}, P_{2}, \ldots, P_{2021}$ be 2021 points in the quarter plane $\{(x, y): x \geq 0, y \geq 0\}$. The centroid of these 2021 points lies at the point $(1,1)$.
Show that there are two distinct points $P_{i}, P_{j}$ such that the distance from $P_{i}$ to $P_{j}$ is no more than $\sqrt{2} / 20$.
Let $n$ be a natural number. Solve in integers the equation
\[x^n + y^n = (x - y)^{n+1}.\]
In the figure, it is given that angle $ C \equal{} 90^{\circ}, \overline{AD} \equal{} \overline{DB}, DE \perp AB, \overline{AB} \equal{} 20$, and $ \overline{AC} \equal{} 12$. The area of quadrilateral $ ADEC$ is:
[asy]unitsize(7);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair A,B,C,D,E;
A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10);
draw(A--B--C--cycle); draw(D--E);
label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE);
draw(rightanglemark(B,D,E,30));[/asy]$ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 58\frac {1}{2} \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 37\frac {1}{2} \qquad\textbf{(E)}\ \text{none of these}$
Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression
$S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.
Let $ABCD$ be a square with side $20$ and $T_1, T_2, ..., T_{2000}$ are points in $ABCD$ such that no $3$ points in the set $S = \{A, B, C, D, T_1, T_2, ..., T_{2000}\}$ are collinear. Prove that there exists a triangle with vertices in $S$, such that the area is less than $1/10$.
Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.
Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.
Inside a triangle ABC, consider points $D, E$ such that $\angle ABD =\angle DBE=\angle EBC$ and $\angle ACD=\angle DC E=\angle ECB$. Calculate angles $\angle BDE$, $\angle B EC$, $\angle D E C$ in terms of the angle of the triangle $ABC$.
Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that
\[f(x+y)\ge f(x)+yf(f(x)) \]
for every $x,y\in (0,\infty )$.