Found problems: 85335
Kvant 2023, M2764
Let $BE{}$ and $CF$ be heights in the acute-angled triangle $ABC{}$ and let $O{}$ be its circumcenter. The points $M{}$ and $N{}$ are selected on the side $BC{}$ so that $BM=CN.{}$ The line $BE{}$ intersects the circle $(MBF)$ a second time at $P{}$ and the line $CF{}$ intersects the circle $(NCE)$ a second time at $Q.{}$ Prove that the lines $PF, QE$ and $AO{}$ intersect at the same point.
[i]Proposed by Luu Dong[/i]
2017 Canadian Open Math Challenge, A3
Source: 2017 Canadian Open Math Challenge, Problem A3
-----
Two $1$ × $1$ squares are removed from a $5$ × $5$ grid as shown.
[asy]
size(3cm);
for(int i = 0; i < 6; ++i) {
for(int j = 0; j < 6; ++j) {
if(j < 5) {
draw((i, j)--(i, j + 1));
}
}
}
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
draw((0,5)--(1,5));
draw((2,5)--(5,5));
draw((0,0)--(2,0));
draw((3,0)--(5,0));
[/asy]
Determine the total number of squares of various sizes on the grid.
2011 Sharygin Geometry Olympiad, 3
Let $ABC$ be a triangle with $\angle{A} = 60^\circ$. The midperpendicular of segment $AB$ meets line $AC$ at point $C_1$. The midperpendicular of segment $AC$ meets line $AB$ at point $B_1$. Prove that line $B_1C_1$ touches the incircle of triangle $ABC$.
2018 Regional Competition For Advanced Students, 4
Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$.
Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$.
[i]Proposed by Richard Henner[/i]
2006 Bulgaria National Olympiad, 2
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$
\[f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}\]
a) Prove that $f(2x)=4f(x)$ for all $x>0$;
b) Find all such functions.
[i]Nikolai Nikolov, Oleg Mushkarov [/i]
2020 AIME Problems, 11
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2015 District Olympiad, 4
Let $ f: (0,\infty)\longrightarrow (0,\infty) $ a non-constant function having the property that $ f\left( x^y\right) = \left( f(x)\right)^{f(y)},\quad\forall x,y>0. $
Show that $ f(xy)=f(x)f(y) $ and $ f(x+y)=f(x)+f(y), $ for all $ x,y>0. $
1996 IMO Shortlist, 5
Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that
\[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}.
\]
2010 Thailand Mathematical Olympiad, 7
Let $a, b, c$ be positive reals. Show that $\frac{a^5}{bc^2} + \frac{b^5}{ca^2} + \frac{c^5}{ab^2} \ge a^2 + b^2 + c^2.$
2014 Contests, 3
The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$. A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$. Tetermine the number of allowed routes.
2016 Turkey Team Selection Test, 2
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have?
LMT Speed Rounds, 2011.2
Julia and Hansol are having a math-off. Currently, Julia has one more than twice as many points as Hansol. If Hansol scores $6$ more points in a row, he will tie Julia’s score. How many points does Julia have?
2011 NIMO Summer Contest, 11
How many ordered pairs of positive integers $(m, n)$ satisfy the system
\begin{align*}
\gcd (m^3, n^2) & = 2^2 \cdot 3^2,
\\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6,
\end{align*}
where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?
2018 China Second Round Olympiad, 1
Let $ a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n,A,B$ are positive reals such that $ a_i\leq b_i,a_i\leq A$ $(i=1,2,\cdots,n)$ and $\frac{b_1 b_2 \cdots b_n}{a_1 a_2 \cdots a_n}\leq \frac{B}{A}.$ Prove that$$\frac{(b_1+1) (b_2+1) \cdots (b_n+1)}{(a_1+1) (a_2+1) \cdots (a_n+1)}\leq \frac{B+1}{A+1}.$$
1992 Poland - First Round, 7
Given are the points $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$ in the space. Let $P_{ij} (i,j \in 0,1,2,3)$ be the point determined by the equality: $\overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j}$. Find the volume of the smallest convex polyhedron which contains all the points $P_{ij}$.
2017 Brazil Team Selection Test, 3
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
[i]Proposed by Evan Chen, Taiwan[/i]
2019 JBMO Shortlist, A3
Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 11 \}$ with $A \cup B = X$. Let
$P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$.
Find the minimum and maximum possible value of $P_A +P_B$ and find all possible equality
cases.
[i]Proposed by Greece[/i]
2019 Taiwan APMO Preliminary Test, P7
Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.
2009 Czech and Slovak Olympiad III A, 6
Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.
2017 Estonia Team Selection Test, 5
The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?
2000 France Team Selection Test, 2
A function from the positive integers to the positive integers satisfies these properties
1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$.
2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$.
Prove that $f(2)=2, f(3)=3, f(1999)=1999$.
2002 Moldova National Olympiad, 4
At least two of the nonnegative real numbers $ a_1,a_2,...,a_n$ aer nonzero. Decide whether $ a$ or $ b$ is larger if
$ a\equal{}\sqrt[2002]{a_1^{2002}\plus{}a_2^{2002}\plus{}\ldots\plus{}a_n^{2002}}$
and
$ b\equal{}\sqrt[2003]{a_1^{2003}\plus{}a_2^{2003}\plus{}\ldots\plus{}a_n^{2003} }$
2019 Brazil Undergrad MO, 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any $(x, y)$ real numbers we have
$f(xf(y)+f(x))+f(y^2)=f(x)+yf(x+y)$
2016 CCA Math Bonanza, T8
As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$.
[i]2016 CCA Math Bonanza Team #8[/i]
1992 Czech And Slovak Olympiad IIIA, 2
Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$