Found problems: 85335
2016 NIMO Problems, 6
Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$.
[i]Proposed by Michael Ren[/i]
2019 USMCA, 2
A [i]trifecta[/i] is an ordered triple of positive integers $(a, b, c)$ with $a < b < c$ such that $a$ divides $b$, $b$ divides $c$, and $c$ divides $ab$. What is the largest possible sum $a + b + c$ over all trifectas of three-digit integers?
1993 National High School Mathematics League, 14
If $0<a<b$, given two fixed points $A(a,0),B(b,0)$. Draw lines $l$ passes $A$, $m$ passes $B$. They have four different intersections with parabola $y^2=x$. If the four points are concyclic, find the path of $P(P=l\cap m)$.
2010 AMC 10, 16
Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter?
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 33 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 37$
2011 China Second Round Olympiad, 6
In a tetrahedral $ABCD$, given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$, $AD=BD=3$, and $CD=2$. Find the radius of the circumsphere of $ABCD$.
2006 Rioplatense Mathematical Olympiad, Level 3, 1
(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$.
(b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that \[\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.\]
2004 Harvard-MIT Mathematics Tournament, 2
A parallelogram has $3$ of its vertices at $(1, 2)$, $(3,8)$, and $(4, 1)$. Compute the sum of the possible $x$-coordinates for the $4$th vertex.
2014 India IMO Training Camp, 3
In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.
1991 Tournament Of Towns, (304) 1
$32$ knights live in a kingdom. Some of them are servants of others. A servant may have only one master and any master is more wealthy than any of his servants. A knight having not less than $4$ servants is called a baron. What is the maximum number of barons? (The kingdom is ruled by the law: “My servant’s servant is not my servant”.
(A. Tolpygo, Kiev)
2017 Yasinsky Geometry Olympiad, 4
In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.
2023 Singapore Junior Math Olympiad, 3
Define a domino to be a $1\times 2$ rectangular block. A $2023\times 2023$ square grid is filled with non-overlapping dominoes, leaving a single $1\times 1$ gap. John then repeatedly slides dominoes into the gap; each domino is moved at most once. What is the maximum number of times that John could have moved a domino? (Example: In the $3\times 3$ grid shown below, John could move 2 dominoes: $D$, followed by $A$.)
[asy]
unitsize(18);
draw((0,0)--(3,0)--(3,3)--(0,3)--(0,0)--cycle);
draw((0,1)--(3,1));
draw((2,0)--(2,3));
draw((1,1)--(1,3));
label("A",(0.5,2));
label("B",(1.5,2));
label("C",(2.5,2));
label("D",(1,0.5));
[/asy]
1971 IMO Longlists, 38
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.
2010 Contests, 2
Find all natural numbers $ n > 1$ such that $ n^{2}$ does $ \text{not}$ divide $ (n \minus{} 2)!$.
2012 Iran MO (3rd Round), 2
Suppose $s,k,t\in \mathbb N$. We've colored each natural number with one of the $k$ colors, such that each color is used infinitely many times. We want to choose a subset $\mathcal A$ of $\mathbb N$ such that it has $t$ disjoint monochromatic $s$-element subsets. What is the minimum number of elements of $A$?
[i]Proposed by Navid Adham[/i]
2023 Ukraine National Mathematical Olympiad, 11.5
Let's call a polynomial [i]mixed[/i] if it has both positive and negative coefficients ($0$ isn't considered positive or negative). Is the product of two mixed polynomials always mixed?
[i]Proposed by Vadym Koval[/i]
Putnam 1938, B1
Do either $(1)$ or $(2)$
$(1)$ Let $A$ be matrix $(a_{ij}), 1 \leq i,j \leq 4.$ Let $d =$ det$(A),$ and let $A_{ij}$ be the cofactor of $a_{ij}$, that is, the determinant of the $3 \times 3$ matrix formed from $A$ by deleting $a_{ij}$ and other elements in the same row and column. Let $B$ be the $4 \times 4$ matrix $(A_{ij})$ and let $D$ be det $B.$ Prove $D = d^3$.
$(2)$ Let $P(x)$ be the quadratic $Ax^2 + Bx + C.$ Suppose that $P(x) = x$ has unequal real roots. Show that the roots are also roots of $P(P(x)) = x.$ Find a quadratic equation for the other two roots of this equation. Hence solve $(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.$
2017 Harvard-MIT Mathematics Tournament, 8
Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.
2020 Polish Junior MO First Round, 7.
Consider the right prism with the rhombus with side $a$ and acute angle $60^{\circ}$ as a base. This prism was intersected by some plane intersecting its side edges, such that the cross-section of the prism and the plane is a square. Determine all possible lengths of the side of this square.
1988 AMC 8, 11
$ \sqrt{164} $ is
$ \text{(A)}\ 42\qquad\text{(B)}\ \text{less than }10\qquad\text{(C)}\ \text{between }10\text{ and }11\qquad\text{(D)}\ \text{between }11\text{ and }12\qquad\text{(E)}\ \text{between }12\text{ and }13 $
2001 Moldova National Olympiad, Problem 5
Consider all quadratic trinomials $x^2+px+q$ with $p,q\in\{1,\ldots,2001\}$. Which of them has more elements: those having integer roots, or those having no real roots?
2016 IOM, 1
Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.
2022 Thailand TSTST, 3
Let $S$ be the set of the positive integers greater than $1$, and let $n$ be from $S$. Does there exist a function $f$ from $S$ to itself such that for all pairwise distinct positive integers $a_1, a_2,...,a_n$ from $S$, we have $f(a_1)f(a_2)...f(a_n)=f(a_1^na_2^n...a_n^n)$?
1985 AMC 12/AHSME, 6
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is
$ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$
2019 Purple Comet Problems, 22
Let $a$ and $b$ positive real numbers such that $(65a^2 + 2ab + b^2)(a^2 + 8ab + 65b^2) = (8a^2 + 39ab + 7b^2)^2$. Then one possible value of $\frac{a}{b}$ satis es $2 \left(\frac{a}{b}\right) = m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2006 China Team Selection Test, 3
Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.