Found problems: 85335
Two distinct points $A$ and $B$ are marked on the left half of the parabola $y = x^2$. Consider any pair of parallel lines which pass through $A$ and $B$ and intersect the right half of the parabola at points $C$ and $D$. Let $K$ be the intersection point of the diagonals $AC$ and $BD$ of the obtained trapezoid $ABCD$. Let $M, N$ be the midpoints of the bases of $ABCD$. Prove that the difference $KM - KN$ depends only on the choice of points $A$ and $B$ but does not depend on the pair of parallel lines described above.
(I. Voronovich)
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overline{DP}$, we obtain a triangular pyramid, all four of whose faces are isosceles triangles. Find the volume of this pyramid.
[asy]
pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
draw(B--C--P--D--C^^D--A);
filldraw(A--P--B--cycle, gray, black);
label("$A$", A, SE);
label("$B$", B, NE);
label("$C$", C, NW);
label("$D$", D, SW);
label("$P$", P, N);
label("$13\sqrt{3}$", A--D, S);
label("$12\sqrt{3}$", A--B, E);[/asy]
Points $A,B,C,D,E,F$ is on the circle $O.$ A line $\ell$ is tangent to $O$ at $E$ is parallel to $AC$ and $DE>EF.$ Let $P,Q$ be the intersection of $\ell$ and $BC,CD$ ,respectively and let $R,S$ be the intersection of $\ell$ and $CF,DF$ ,respectively. Show that $PQ=RS$ if and only if $QE=ER.$
Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$, $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b$ and $c$ having the same tens digit (first digit)?
By composing a symmetry of axis $r$ with a right angle rotation around from a point $P$ that does not belong to the line, another movement $M$ results. Is $M$ an axis symmetry? Is there any line invariant through $M$?
The diagonals of an cyclic quadrilateral $ABCD$ intersect at $O$. The circumcircles of triangle $AOB$ and $COD$ intersect lines $BC$ and $AD$, for the second time, at points $M, N, P$and $Q$. Prove that the $MNPQ$ quadrilateral is inscribed in a circle of center $O$.
Two points $ B$ and $ C$ are in a plane. Let $ S$ be the set of all points $ A$ in the plane for which $ \triangle ABC$ has area $ 1$. Which of the following describes $ S$?
$ \textbf{(A)}\ \text{two parallel lines}\qquad
\textbf{(B)}\ \text{a parabola}\qquad
\textbf{(C)}\ \text{a circle}\qquad
\textbf{(D)}\ \text{a line segment}\qquad
\textbf{(E)}\ \text{two points}$
For the real numbers $a_1,...,a_n, b_1,...,b_m$ , the following relations hold:
1) $|a_i|= |b_j|=1$, $i=1,...,n$ ,$j=1,...,m$
2) $a_1\sqrt{2+a_2\sqrt{2+...+a_n\sqrt2}}=b_1\sqrt{2+b_2\sqrt{2+...+b_m\sqrt2}}$
Prove that $n = m$ and $a_i=b_i$ , $i=1,...,n$
\[\int_{-1}^1 x^{2022}\cos\left(\tfrac \pi {12}-x\right)\sin\left(\tfrac \pi{12}+x\right)\,\mathrm dx\]
[i]Proposed by Michael Duncan, Connor Gordon, and Vlad Oleksenko[/i]
Let $X$ be a set containing $n$ elements. Find the number of ordered triples $(A,B, C)$ of subsets of $X$ such that $A$ is a subset of $B$ and $B$ is a proper subset of $C$.
Let $X$ be a set of $100$ elements. Find the smallest possible $n$ satisfying the following condition: Given a sequence of $n$ subsets of $X$, $A_1,A_2,\ldots,A_n$, there exists $1 \leq i < j < k \leq n$ such that
$$A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k.$$
Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that,
\[
\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}
\]
where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.
Let $f:[0,1]\to[0,1]$ be a differentiable function such that $|f'(x)|\ne1$ for all $x\in[0,1]$. Prove that there exist unique $\alpha,\beta\in[0,1]$ such that $f(\alpha)=\alpha$ and $f(\beta)=1-\beta$.
Find whether the number of powers of 2, which have a digit sum smaller than $2019^{2019}$, is finite or infinite.
The number of real solutions of the equation \[|x-2|+|x-3|=1\] is
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }2\qquad
\textbf{(D) }3\qquad
\textbf{(E) }\text{more than }3\qquad$
Show that $ \frac{\frac{1}{1\cdot 2} +\frac{1}{3\cdot 4}+\cdots +\frac1{1997\cdot 1998}}{\frac{2}{1000\cdot 1998} +\frac{1}{1001\cdot 1997}} $ is an integer number.
[i]Bogdan Enescu[/i]
The sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=1, \; a_{2}=12, \; a_{3}=20, \; a_{n+3}= 2a_{n+2}+2a_{n+1}-a_{n}.\] Prove that $1+4a_{n}a_{n+1}$ is a square for all $n \in \mathbb{N}$.
Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo:
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let $F_A$ be the (associated with $\Omega$) Feuerbach point of the triangle formed by the tangents to $\Omega$ at $B,C,D$, that is, the point of tangency of $\Omega$ and the nine-point circle of that triangle. Define $F_B, F_C, F_D$ similarly. Let $A'$ be the intersection of the tangents to $\Omega$ at $A$ and $F_A$. Define $B', C', D'$ similarly.
Prove that quadrilaterals $ABCD$ and $A'B'C'D'$ are similar
Let $x,y,z$ be positive real numbers satisfying $4x^2 - 2xy + y^2 = 64, y^2 - 3yz +3z^2 = 36,$ and $4x^2 +3z^2 = 49.$ If the maximum possible value of $2xy +yz -4zx$ can be expressed as $\sqrt{n}$ for some positive integer $n,$ find $n.$
Prove that there are exactly $2(2^{n-1}-1)$ ways of dealing $n$ cards to two persons. (The persons may receive unequal numbers of cards.)
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops?
$\textbf{(A) }\frac{9}{16}\qquad\textbf{(B) }\frac{5}{8}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{25}{32}\qquad\textbf{(E) }\frac{13}{16}$
Given an equilateral $\triangle ABC$, find the locus of points $P$ such that $\angle APB=\angle BPC$.
Consider a $2\times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below:
$$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$$
Given a positive integer $n \ge 2$, we define $f(n)$ to be the sum of all remainders obtained by dividing $n$ by all positive integers less than $n$. For example dividing $5$ with $1, 2, 3$ and $4$ we have remainders equal to $0, 1, 2$ and $1$ respectively. Therefore $f(5) = 0 + 1 + 2 + 1 = 4$. Find all positive integers $n \ge 3$ such that $f(n) = f(n - 1) + (n - 2)$.