Found problems: 85335
1994 IMC, 1
a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$.
b) How many zero elements are there in the inverse of the $n\times n$ matrix
$$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\
1&2&2&2&\ldots&2\\
1&2&1&1&\ldots&1\\
1&2&1&2&\ldots&2\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
1&2&1&2&\ldots&\ddots
\end{pmatrix}$$
2007 Mexico National Olympiad, 2
Given an equilateral $\triangle ABC$, find the locus of points $P$ such that $\angle APB=\angle BPC$.
2018 PUMaC Algebra A, 5
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
2014 AMC 12/AHSME, 5
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?
[asy]
fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey);
for(int i = 0; i < 4; ++i){
for(int j = 0; j < 2; ++j){
fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white);
}
}[/asy]
$\textbf{(A) }26\qquad\textbf{(B) }28\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
2023 Chile Junior Math Olympiad, 3
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$.
[img]https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.png[/img]
PS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3[/url]
2014 Sharygin Geometry Olympiad, 8
Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.
(Tran Quang Hung)
2016 Postal Coaching, 5
For even positive integer $n$ we put all numbers $1, 2, \cdots , n^2$ into the squares of an $n \times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that it is possible to have $\frac{S_1}{S_2}=\frac{39}{64}$.
1995 Korea National Olympiad, Problem 1
For any positive integer $m$,show that there exist integers $a,b$ satisfying
$\left | a \right |\leq m$, $ \left | b \right |\leq m$, $0< a+b\sqrt{2}\leq \frac{1+\sqrt{2}}{m+2}$
2019 European Mathematical Cup, 1
Every positive integer is marked with a number from the set $\{ 0,1,2\}$, according to the following rule:
$$\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0.$$
Let $S$ denote the sum of marks of the first $2019$ positive integers. Determine the maximum possible value of $S$.
[i]Proposed by Ivan Novak[/i]
2023 CMIMC Integration Bee, 4
\[\int_0^\infty x e^{-\sqrt[3]{x}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2009 Sharygin Geometry Olympiad, 7
Given points $O, A_1, A_2, ..., A_n$ on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.
(A.Glazyrin)
2019 BAMO, E/3
In triangle $\vartriangle ABC$, we have marked points $A_1$ on side $BC, B_1$ on side $AC$, and $C_1$ on side $AB$ so that $AA_1$ is an altitude, $BB_1$ is a median, and $CC_1$ is an angle bisector. It is known that $\vartriangle A_1B_1C_1$ is equilateral. Prove that $\vartriangle ABC$ is equilateral too.
(Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $BB_1$ we know that $B_1$ is the midpoint of side $AC$ in $\vartriangle ABC$.)
2017 Dutch BxMO TST, 1
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.
May Olympiad L1 - geometry, 2008.4
Let $ABF$ be a right-angled triangle with $\angle AFB = 90$, a square $ABCD$ is externally to the triangle. If $FA = 6$, $FB = 8$ and $E$ is the circumcenter of the square $ABCD$, determine the value of $EF$
2003 IMO Shortlist, 2
Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P6
For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$.
[i]Proposed by Mirko Petrusevski[/i]
2008 Harvard-MIT Mathematics Tournament, 3
Farmer John has $ 5$ cows, $ 4$ pigs, and $ 7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)
1998 Croatia National Olympiad, Problem 4
Among any $79$ consecutive natural numbers there exists one whose sum of digits is divisible by $13$. Find a sequence of $78$ consecutive natural numbers for which the above statement fails.
2012 Dutch IMO TST, 2
There are two boxes containing balls. One of them contains $m$ balls, and the other contains $n$ balls, where $m, n > 0$. Two actions are permitted:
(i) Remove an equal number of balls from both boxes.
(ii) Increase the number of balls in one of the boxes by a factor $k$.
Is it possible to remove all of the balls from both boxes with just these two actions,
1. if $k = 2$?
2. if $k = 3$?
1961 AMC 12/AHSME, 5
Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:
${{ \textbf{(A)}\ (x-2)^4 \qquad\textbf{(B)}\ (x-1)^4 \qquad\textbf{(C)}\ x^4 \qquad\textbf{(D)}\ (x+1)^4 }\qquad\textbf{(E)}\ x^4+1} $
1997 APMO, 1
Given:
\[ S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}} \]
where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. $k=\frac{n(n+1)}{2}$ for $n = 1$, $2$, $\ldots$,$1996$). Prove that $S>1001$.
2021 LMT Spring, B6
Maisy is at the origin of the coordinate plane. On her first step, she moves $1$ unit up. On her second step, she moves $ 1$ unit to the right. On her third step, she moves $2$ units up. On her fourth step, she moves $2$ units to the right. She repeats this pattern with each odd-numbered step being $ 1$ unit more than the previous step. Given that the point that Maisy lands on after her $21$st step can be written in the form $(x, y)$, find the value of $x + y$.
Proposed by Audrey Chun
2004 India Regional Mathematical Olympiad, 1
Consider in the plane a circle $\Gamma$ with centre O and a line l not intersecting the circle. Prove that there is a unique point Q on the perpendicular drawn from O to line l, such that for any point P on the line l, PQ represents the length of the tangent from P to the given circle.
2010 Costa Rica - Final Round, 4
Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]
2009 Croatia Team Selection Test, 4
Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.