Found problems: 85335
2016 SDMO (High School), 3
Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.
2010 ELMO Shortlist, 6
Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic.
[i]Carl Lian.[/i]
1995 Yugoslav Team Selection Test, Problem 2
A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.
2022 Caucasus Mathematical Olympiad, 8
Paul can write polynomial $(x+1)^n$, expand and simplify it, and after that change every coefficient by its reciprocal. For example if $n=3$ Paul gets $(x+1)^3=x^3+3x^2+3x+1$ and then $x^3+\frac13x^2+\frac13x+1$. Prove that Paul can choose $n$ for which the sum of Paul’s polynomial coefficients is less than $2.022$.
MMPC Part II 1996 - 2019, 1997
[b]p1.[/b] It can be shown in Calculus that the area between the x-axis and the parabola $y=kx^2$ (к is a positive constant) on the $x$-interval $0 \le x \le a$ is $\frac{ka^3}{3}$
a) Find the area between the parabola $y=4x^2$ and the x-axis for $0 \le x \le 3$.
b) Find the area between the parabola $y=5x^2$ and the x-axis for $-2 \le x \le 4$.
c) A square $2$ by $2$ dartboard is situated in the $xy$-plane with its center at the origin and its sides parallel to the coordinate axes. Darts that are thrown land randomly on the dartboard. Find the probability that a dart will land at a point of the dartboard that is nearer to the point $(0, 1)$ than to the bottom edge of the dartboard.
[b]p2.[/b] When two rows of a determinant are interchanged, the value of the determinant changes sign. There are also certain operations which can be performed on a determinant which leave its value unchanged. Two such operations are changing any row by adding a constant multiple of another row to it, and changing any column by adding a constant multiple of another column to it. Often these operations are used to generate lots of zeroes in a determinant in order to simplify computations. In fact, if we can generate zeroes everywhere below the main diagonal in a determinant, the value of the determinant is just the product of all the entries on that main diagonal. For example, given the determinant $\begin{vmatrix} 1 & 2 & 3 \\
2 & 6 & 2 \\
3 & 10 & 4
\end{vmatrix}$ we add $-2$ times the first row to the second row, then add $-2$ times the second row to the third row, giving the new determinant $\begin{vmatrix} 1 & 2 & 3 \\
0 & 2 & -4 \\
0 & 0 & 3
\end{vmatrix}$ , and the value is the product of the diagonal entries: $6$.
a) Transform this determinant into another determinant with zeroes everywhere below the main diagonal, and find its value: $\begin{vmatrix} 1 & 3 & -1 \\
4 & 7 & 2 \\
3 & -6 & 5
\end{vmatrix}$
b) Do the same for this determinant: $\begin{vmatrix} 0 & 1 & 2 & 3 \\
1 & 0 & 1 & 2 \\
2 & 1 & 0 & 1 \\
3 & 2 & 1 & 0
\end{vmatrix}$
[b]p3.[/b] In Pascal’s triangle, the entries at the ends of each row are both $1$, and otherwise each entry is the sum of the two entries diagonally above it:
Row Number
$0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1$
$1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\,\,1$
$2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1 \,\, 2 \,\,1$
$3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1\,\, 3 \,\, 3 \,\, 1$
$4\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 \,\,4 \,\, 6 \,\, 4 \,\, 1$
$...\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...$
This triangle gives the binomial coefficients in expansions like $( a + b)^3 = 1a^3 + 3a^2 b + 3 ab^2 + 1b^3$ .
a) What is the sum of the numbers in row #$5$ of Pascal's triangle?
b) What is the sum of the numbers in row #$n$ of Pascal's triangle?
c) Show that in row #$6$ of Pascal's triangle, the sum of all the numbers is exactly twice the sum of the first, third, fifth, and seventh numbers in the row.
d) Prove that in row #$n$ of Pascal's triangle, the sum of ail the numbers is exactly twice the sum of the numbers in the odd positions of that row.
[b]p4.[/b] The product: of several terms is sometimes described using the symbol $\Pi$ which is capital pi, the Greek equivalent of $p$, for the word "product". For example the symbol $\prod^4_{k=1}(2k +1)$ means the product of numbers of the form $(2k + 1)$, for $k=1,2,3,4$. Thus it equals $945$.
a) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k}{k + 2}$
b) Evaluate as a reduced fraction $\prod_{k=1}^{10} \frac{k^2 + 10k+ 17}{k^2+4k + 41}$
c) Evaluate as a reduced fraction $\prod_{k=1}^{\infty}\frac{k^3-1}{k^3+1}$
[b]p5.[/b] a) In right triangle $CAB$, the median $AF$, the angle bisector $AE$, and the altitude $AD$ divide the right angld $A$ into four equal angles. If $AB = 1$, find the area of triangle $AFE$.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/0d4a83e58a65c2546ce25d1081b99d45e30729.png[/img]
b) If in any triangle, an angle is divided into four equal angles by the median, angle bisector, and altitude drawn from that angle, prove that the angle must be a right angle.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1990 Nordic, 2
Let $a_1, a_2, . . . , a_n$ be real numbers. Prove
$\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le \sqrt{a_1^2+ a_2^2+ . . . + a_n^2} $ (1)
When does equality hold in (1)?
1997 Bundeswettbewerb Mathematik, 1
Three faces of a regular tetrahedron are painted in white and the remaining one in black. Initially, the tetrahedron is positioned on a plane with the black face down. It is then tilted several times over its edges. After a while it returns to its original position. Can it now have a white face down?
2020 Malaysia IMONST 2, 3
Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.
2016 Korea Winter Program Practice Test, 4
Let $x,y,z \ge 0$ be real numbers such that $(x+y-1)^2+(y+z-1)^2+(z+x-1)^2=27$.
Find the maximum and minimum of $x^4+y^4+z^4$
2023 LMT Fall, 4
The numbers $1$, $2$, $3$, and $4$ are randomly arranged in a $2$ by $2$ grid with one number in each cell. Find the probability the sum of two numbers in the top row of the grid is even.
[i]Proposed by Muztaba Syed and Derek Zhao[/i]
[hide=Solution]
[i]Solution. [/i]$\boxed{\dfrac{1}{3}}$
Pick a number for the top-left. There is one number that makes the sum even no matter what we pick. Therefore, the
answer is $\boxed{\dfrac{1}{3}}$.[/hide]
2003 National Olympiad First Round, 31
Positive integers are written into squares of a infinite chessboard such that a number $n$ is written $n$ times. If the absolute differences of numbers written into any two squares having a common side is not greater than $k$, what is the least possible value of $k$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Geometry Mathley 2011-12, 15.1
Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$.
Nguyễn Minh Hà
2015 USAMTS Problems, 3
Let $P$ be a convex n-gon in the plane with vertices labeled $V_1,...,V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles the blue and green regions are the same. Suppose $P$ has exactly one balancing point/ Show that the balancing point must be a vertex of $P$
2005 Slovenia National Olympiad, Problem 4
The friends Alex, Ben, and Charles prepared a lot of labels and wrote one of the numbers $2,3,4,5,6,7,8$ on each label. Then Mary joined them and glued one label onto the forehead of each friend. Of course, each of the friends can see the labels on the others’ foreheads, but not the one on his own forehead. Mary told them: ”The numbers on your foreheads are not all distinct, and their product is a perfect square.” Can any of the friends find out the number on his forehead?
2019 Saudi Arabia JBMO TST, 2
Let $a, b, c$ be positive real numbers. Prove that
$$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$
1993 Romania Team Selection Test, 4
For each integer $n > 3$ find all quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $n_1 +n_2 +n_3 +n_4 = n$ which maximize the expression $$\frac{n!}{n_1!n_2!n_3!n_4!}2^{ {n_1 \choose 2}+{n_2 \choose 2}+{n_3 \choose 2}+{n_4 \choose 2}+n_1n_2+n_2n_3+n_3n_4}$$
2000 Stanford Mathematics Tournament, 19
Eleven pirates find a treasure chest. When they split up the coins in it, they find that there are 5 coins left. They throw one pirate overboard and split the coins again, only to find that there are 3 coins left over. So, they throw another pirate over and try again. This time, the coins split evenly.
What is the least number of coins there could have been?
2008 Balkan MO Shortlist, G7
In the non-isosceles triangle $ABC$ consider the points $X$ on $[AB]$ and $Y$ on $[AC]$ such that $[BX]=[CY]$, $M$ and $N$ are the midpoints of the segments $[BC]$, respectively $[XY]$, and the straight lines $XY$ and $BC$ meet in $K$. Prove that the circumcircle of triangle $KMN$ contains a point, different from $M$ , which is independent of the position of the points $X$ and $Y$.
2025 Romanian Master of Mathematics, 1
Let $n > 10$ be an integer, and let $A_1, A_2, \dots, A_n$ be distinct points in the plane such that the distances between the points are pairwise different. Define $f_{10}(j, k)$ to be the 10th smallest of the distances from $A_j$ to $A_1, A_2, \dots, A_k$, excluding $A_j$ if $k \geq j$. Suppose that for all $j$ and $k$ satisfying $11 \leq j \leq k \leq n$, we have $f_{10}(j, j - 1) \geq f_{10}(k, j - 1)$. Prove that $f_{10}(j, n) \geq \frac{1}{2} f_{10}(n, n)$ for all $j$ in the range $1 \leq j \leq n - 1$.
[i]Proposed by Morteza Saghafian, Iran[/i]
2017 International Olympic Revenge, 2
A polynomial is [i]good[/i] if it has integer coefficients, it is monic, all its roots are distinct, and there exists a disk with radius $0.99$ on the complex plane that contains all the roots. Prove that there is no [i]good[/i] polynomial for a sufficient large degree.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil.[/i]
2005 China Team Selection Test, 1
Triangle $ABC$ is inscribed in circle $\omega$. Circle $\gamma$ is tangent to $AB$ and $AC$ at points $P$ and $Q$ respectively. Also circle $\gamma$ is tangent to circle $\omega$ at point $S$. Let the intesection of $AS$ and $PQ$ be $T$. Prove that $\angle{BTP}=\angle{CTQ}$.
2023 Bulgaria JBMO TST, 3
Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.
2017 Brazil National Olympiad, 6.
[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.
2016 Czech-Polish-Slovak Junior Match, 3
On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer.
Poland
2012 China Team Selection Test, 2
Find all integers $k\ge 3$ with the following property: There exist integers $m,n$ such that $1<m<k$, $1<n<k$, $\gcd (m,k)=\gcd (n,k) =1$, $m+n>k$ and $k\mid (m-1)(n-1)$.