Found problems: 85335
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.
Let $\{a_n\}$ be the sequence such that $a_0=2019$ and $$a_n=-\frac{2020}{n}\sum_{k=0}^{n-1}a_k.$$ Compute the last three digits of $\sum_{n=1}^{2020}2020^na_nn$.
Points $K$ and $L$ on the sides $AB$ and $BC$ of parallelogram $ABCD$ are such that $\angle AKD = \angle CLD$. Prove that the circumcenter of triangle $BKL$ is equidistant from $A$ and $C$.
[i]Proposed by I.I.Bogdanov[/i]
Given a sequence $(a_n)$, with $a_1 = 4$ and $a_{n+1} = a_n^2-2 (\forall n \in\mathbb{N})$, prove that there is a triangle with side lengths $a_{n-1}, a_n, a_{n+1},$ and that its area is equal to an integer.
In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.
Determine all ordered quadruples of real numbers $(x_1, x_2, x_3, x_4)$ for which the following system of equations exists, is fulfilled:
$$x_1x_2 + x_1x_3 + x_2x_3 + x_4 = 2$$
$$x_1x_2 + x_1x_4 + x_2x_4 + x_3 = 2$$
$$x_1x_3 + x_1x_4 + x_3x_4 + x_2 = 2$$
$$x_2x_3 + x_2x_4 + x_3x_4 + x_1 = 2$$
Show that $ \det \left( I_n+A \right)\ge 1, $ for any $ n\times n $ antisymmetric real matrix $ A. $
A (not necessarily regular) tetrahedron $A_1A_2A_3A_4$ is given in space. For each pair of indices $1\leq i<j\leq 4$, an ellipsoid with foci $A_i,A_j$ and string length $\ell_{ij}$, for positive numbers $\ell_{ij}$, is given (in all 6 ellipsoids were built).
For each $i=1,2$, a pair of points $X_i\neq X'_i$ was chosen so that $X_i, X'_i$ both belong to all three ellipsoids with $A_i$ as one of their foci. Prove that the lines $X_1X'_1, X_2X'_2$ share a point in space if and only if
\[\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23}\]
[i]Remark: An [u]ellipsoid[/u] with foci $P,Q$ and string length $\ell>|PQ|$ is defined here as the set of points $X$ in space for which $|XQ|+|XP|=\ell$.[/i]
$12$ distinct points are equally spaced around a circle. How many ways can Bryan choose $3$ points (not in any order) out of these $12$ points such that they form an acute triangle (Rotations of a set of points are considered distinct).
[i]Proposed by Bryan Guo [/i]
Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$.
After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam?
[i]Proposed by Eric Wang[/i]
Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?
- Meat: beef, chicken, pork
- Vegetables: baked beans, corn, potatoes, tomatoes
- Dessert: brownies, chocolate cake, chocolate pudding, ice cream
$ \text{(A)}\ 4\qquad\text{(B)}\ 24\qquad\text{(C)}\ 72\qquad\text{(D)}\ 80\qquad\text{(E)}\ 144 $
A circle intersects equilateral triangle $\vartriangle XY Z$ at $A,$ $B$, $C$, $D$, $E$, and $F$ such that points $X$, $A$, $B$, $Y$ , $C$, $D$, $Z$, $E$, and $F$ lie on the equilateral triangle in that order. If $AC^2 +CE^2 +EA^2 = 1900$ and $BD^2 + DF^2 + FB^2 = 2092$, compute the positive difference between the areas of triangles $\vartriangle ACE$ and $\vartriangle BDF$.
A natural number $n$ is given and written in a row of $n$ numbers, each of which is equal to $0$ or $1$. Then $n - 1$ numbers are written below in a row - one number under each pair of adjacent numbers of the first row. At the same time, $0$ is written under a pair of identical numbers. and under a pair of different ones $1$. Then, under the second row, the third of $n- 2$ numbers is similarly written, etc., until we get a triangular table with $n$ rows. For a given $n$, find the largest possible number of units in such a table.
Let $ABC$ be a triangle with circumcircle $\Gamma$ and circumcentre $O$. The circle with centre on the line $AB$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $D$. Similarly, the circle with centre on the line $AC$ and passing through the points $A$ and $O$ intersects $\Gamma$ again in $E$. Prove that $BD$ is parallel with $CE$.
Outside of the trapezoid $ABCD$ with the smaller base $AB$ are constructed the squares $ADEF$ and $BCGH$. Prove that the perpendicular bisector of $AB$ passes through the midpoint of $FH$.
Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.
Let points $P$ and $Q$ be isogonally conjugated with respect to a triangle $ABC$. The line $PQ$ meets the circumcircle of $ABC$ at point $X$. The reflection of $BC$ about $PQ$ meets $AX$ at point $E$. Prove that $A, P, Q, E$ are concyclic.
$N$ is an integer whose representation in base $b$ is $777$. Find the smallest integer $b$ for which $N$ is the fourth power of an integer.
What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?
(A Shapovalov)
The smallest positive integer which can be expressed as sum of positive perfect cubes (possibly with repetition and/or with a single element sum) in at least two different ways in
$$(A) 8$$
$$(B) 1729$$
$$(C) 2023$$
$$(D) 2024$$
In the triangle $ABC$, points $D, E, F$ are on the sides $BC, CA$ and $AB$ respectively such that $FE$ is parallel to $BC$ and $DF$ is parallel to $CA$, Let P be the intersection of $BE$ and $DF$, and $Q$ the intersection of $FE$ and $AD$. Prove that $PQ$ is parallel to $AB$.
For all positive integers $n$ define $a_n=2 \underbrace{33...3}_{n \, times}$, where digit $3$ occurs $n$ times.
Show that the number $a_{2009}$ has infinitely many multiples in the set $\{a_n | n \in N*\}$.