Found problems: 85335
VI Soros Olympiad 1999 - 2000 (Russia), 9.10
Let $x, y, z$ be real numbers from interval $(0, 1)$. Prove that
$$\frac{1}{x(1-y)}+\frac{1}{y(1-x)}+\frac{1}{z(1-x)}\ge \frac{3}{xyz+(1-x)(1-y)(1-z)}$$
2015 Irish Math Olympiad, 4
Two circles $C_1$ and $C_2$, with centres at $D$ and $E$ respectively, touch at $B$. The circle having $DE$ as diameter intersects the circle $C_1$ at $H$ and the circle $C_2$ at $K$. The points $H$ and $K$ both lie on the same side of the line $DE$. $HK$ extended in both directions meets the circle $C_1$ at $L$ and meets the circle $C_2$ at $M$. Prove that
(a) $|LH| = |KM|$
(b) the line through $B$ perpendicular to $DE$ bisects $HK$.
2016 CMIMC, 3
We have 7 buckets labelled 0-6. Initially bucket 0 is empty, while bucket $n$ (for each $1 \leq n \leq 6$) contains the list $[1,2, \ldots, n]$. Consider the following program: choose a subset $S$ of $[1,2,\ldots,6]$ uniformly at random, and replace the contents of bucket $|S|$ with $S$. Let $\tfrac{p}{q}$ be the probability that bucket 5 still contains $[1,2, \ldots, 5]$ after two executions of this program, where $p,q$ are positive coprime integers. Find $p$.
2016 USA Team Selection Test, 1
Let $\sqrt 3 = 1.b_1b_2b_3 \dots _{(2)}$ be the binary representation of $\sqrt 3$. Prove that for any positive integer $n$, at least one of the digits $b_n$, $b_{n+1}$, $\dots$, $b_{2n}$ equals $1$.
2024 Brazil EGMO TST, 3
Let \( ABC \) be an acute scalene triangle with orthocenter \( H \), and consider \( M \) to be the midpoint of side \( BC \). Define \( P \neq A \) as the intersection point of the circle with diameter \( AH \) and the circumcircle of triangle \( ABC \), and let \( Q \) be the intersection of \( AP \) with \( BC \). Let \( G \neq M \) be the intersection of the circumcircle of triangle \( MPQ \) with the circumcircle of triangle \( AHM \). Show that \( G \) lies on the circle that passes through the feet of the altitudes of triangle \( ABC \).
2001 IMO Shortlist, 3
Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
2023 Princeton University Math Competition, A1 / B3
Let p>3 be a prime and k>0 an integer. Find the multiplicity of X-1 in the factorization of
$ f(X)= X^{p^k-1}+X^{p^k-2}+\cdots+X+1$
modulo p;
in other words, find the unique non-negative integer r such that $ (X - 1)^r $ divides f(X)
\modulo p, but$ (X - 1)^{r+1} $does not divide f(X) \modulo p.
2018 Romania Team Selection Tests, 4
Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.
2002 AMC 10, 15
The digits $ 1$, $ 2$, $ 3$, $ 4$, $ 5$, $ 6$, $ 7$, and $ 9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?
$ \text{(A)}\ 150 \qquad
\text{(B)}\ 160 \qquad
\text{(C)}\ 170 \qquad
\text{(D)}\ 180 \qquad
\text{(E)}\ 190$
1996 Argentina National Olympiad, 6
In a tennis tournament of $10$ players, everyone played against everyone once. In this tournament, if player $i$ won the match against player $j$, then the total number of matches $i$ lost plus the total number of matches $j$ won is greater than or equal to $8$. We will say that three players $i$, $j$, $k$ form an [i]atypical tri[/i]o if $i$ beat $j$, $j$ beat $k$ and $k$ beat $i$. Prove that in the tournament there were exactly $40$ atypical trios.
2002 Austria Beginners' Competition, 4
In a trapezoid $ABCD$ with base $AB$ let $E$ be the midpoint of side $AD$. Suppose further that $2CD=EC=BC=b$. Let $\angle ECB=120^{\circ}$. Construct the trapezoid and determine its area based on $b$.
2007 Junior Macedonian Mathematical Olympiad, 4
The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions:
$a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$
$a_{1} + a_{2} = 20$
$a_{3} + a_{4} + ... + a_{20} \le 20$ .
What is maximum value of the expression:
$a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ?
For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?
1999 Bosnia and Herzegovina Team Selection Test, 4
Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$
2017 Taiwan TST Round 2, 1
Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$
[i]Proposed by William Chao[/i]
1999 Singapore Team Selection Test, 2
Find all possible values of $$ \lfloor \frac{x - p}{p} \rfloor + \lfloor \frac{-x-1}{p} \rfloor $$ where $x$ is a real number and $p$ is a nonzero integer.
Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.
1974 IMO Longlists, 18
Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.
2017 AMC 12/AHSME, 19
Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$
1936 Moscow Mathematical Olympiad, 022
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.
2004 Regional Olympiad - Republic of Srpska, 1
Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares
of two positive integers.
2006 Indonesia MO, 6
Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?
2014 PUMaC Combinatorics A, 6
Let $f(n)$ be the number of points of intersection of diagonals of a $n$-dimensional hypercube that is not the vertex of the cube. For example, $f(3) = 7$ because the intersection points of a cube’s diagonals are at the centers of each face and the center of the cube. Find $f(5)$.
2023 Indonesia TST, N
Let $p,q,r$ be primes such that for all positive integer $n$,
$$n^{pqr}\equiv n (\mod{pqr})$$
Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$
1993 National High School Mathematics League, 2
$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is
$\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure
1992 Baltic Way, 18
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
2004 Romania National Olympiad, 2
The sidelengths of a triangle are $a,b,c$.
(a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$.
(b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.