Found problems: 85335
2004 National High School Mathematics League, 14
Three points $A\left(0,\frac{4}{3}\right),B(-1,0),C(1,0)$ are given. The distance from $P$ to line $BC$ is the geometric mean of that from $P$ to lines $AB$ and $AC$.
[b](a)[/b] Find the path equation of point $P$.
[b](b)[/b] If line $L$ passes $D$ ($D$ is the incenter of $\triangle ABC$ ), and it has three common points with the path of $P$, find the range value of slope $k$ of line $L$.
2016 Iran Team Selection Test, 5
Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.
Russian TST 2020, P3
A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities
$$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$
Prove that there exists a polynomial $F(t)$ in one variable such that
$$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$
1986 National High School Mathematics League, 2
In acute triangle $ABC$, $D\in BC,E\in CA,F\in AB$. Prove that the necessary and sufficient condition of $AD,BE,CF$ are heights of $\triangle ABC$ is that $S=\frac{R}{2}(EF+FD+DE)$.
Note: $S$ is the area of $\triangle ABC$, $R$ is the circumradius of $\triangle ABC$.
2024 Oral Moscow Geometry Olympiad, 5
From point $D$ of parallelogram $ABCD$ were drawn an arbitrary line $\ell_1$, intersecting the segment $AB$ and the line $BC$ at points $C_1$ and $A_1$, respectively, and an arbitrary line $\ell_2$ intersecting the segment $BC$ and the line $AB$ at the points $A_2$ and $C_2$, respectively. Find the locus of the intersection points of the circles $(A_1BC_2)$ and $(A_2BC_1)$ (other than point $B$).
1971 Poland - Second Round, 1
In how many ways can you choose $ k $ squares on a chessboard $ n \times n $ ( $ k \leq n $) so that no two of the chosen squares lie in the same row or column?
MBMT Team Rounds, 2015 E13
A blind ant is walking on the coordinate plane. It is trying to reach an anthill, placed at all points where both the $x$-coordinate and $y$-coordinate are odd. The ant starts at the origin, and each minute it moves one unit either up, down, to the right, or to the left, each with probability $\frac{1}{4}$. The ant moves $3$ times and doesn't reach an anthill during this time. On average, how many additional moves will the ant need to reach an anthill? (Compute the expected number of additional moves needed.)
2012 All-Russian Olympiad, 1
Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have
\[P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)\]
Prove that the polynomial $P(x)$ has at least one real root.
1971 AMC 12/AHSME, 12
For each integer $N>1$, there is a mathematical system in which two or more positive integers are defined to be congruent if they leave the same non-negative remainder when divided by $N$. If $69,90,$ and $125$ are congruent in one such system, then in that same system, $81$ is congruent to
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad \textbf{(E) }8$
2022 CMWMC, R1
[u]Set 1
[/u]
[b]1.1[/b] Compute the number of real numbers x such that the sequence $x$, $x^2$, $x^3$,$ x^4$, $x^5$, $...$ eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence $1$, $2$, $3$, $4$, $5$, $6$, $5$, $6$, $5$, $6$, $...$ is eventually repeating with repeating block $5$, $6$.)
[b]1.2[/b] Let $T$ be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply $a$ by $b$, starts by multiplying $a$ by $b$, but then multiplies that product by b again, and then adds $b$ to the result. Nicole inputs the computation “$k \times k$” into the calculator for some real number $k$ and gets an answer of $10T$. If she instead used a working calculator, what answer should she have gotten?
[b]1.3[/b] Let $T$ be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as $x^2 + y^2$ for integers $x, y$ satisfying $\sqrt{T} \le x \le T$ and $\sqrt{T} \le y \le T$.
PS. You should use hide for answers.
2007 Stanford Mathematics Tournament, 4
What is the area of the smallest triangle with all side lengths rational and all vertices lattice points?
2011 Indonesia TST, 2
Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$
2007 Princeton University Math Competition, 9
There are four spheres each of radius $1$ whose centers form a triangular pyramid where each side has length $2$. There is a 5th sphere which touches all four other spheres and has radius less than $1$. What is its radius?
2023 HMNT, 33
Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Suppose the following three conditions hold:
[list]
[*]The smallest of a common internal tangent of $\omega_1$ and $\omega_2$ is equal to $19.$
[*]The length of a common external tangent of $\omega_1$ and $\omega_2$ is equal to $37.$
[*]If two points $X$ and $Y$ are selected on $\omega_1$ and $\omega_2,$ respectively, uniformly at random, then the expected value of $XY^2$ is $2023.$
[/list]
Compute the distance between the centers of $\omega_1$ and $\omega_2.$
2006 Korea National Olympiad, 6
Prove that for any positive real numbers $x,y$ and $z,$
$xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$
2023 HMNT, 20
Let $ABCD$ be a square of side length $10.$ Point $E$ is on ray $\overrightarrow{AB}$ such that $AE=17,$ and point $F$ is on ray $\overrightarrow{AD}$ such that $AF=14.$ The line through $B$ parallel to $CE$ and the line through $D$ parallel to $CF$ meet at $P.$ Compute the area of quadrilateral $AEPF.$
2023 Princeton University Math Competition, 10
10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form
$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$
where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.
2011 Lusophon Mathematical Olympiad, 3
Consider a sequence of equilateral triangles $T_{n}$ as represented below:
[asy]
defaultpen(linewidth(0.8));size(350);
real r=sqrt(3);
path p=origin--(2,0)--(1,sqrt(3))--cycle;
int i,j,k;
for(i=1; i<5; i=i+1) {
for(j=0; j<i; j=j+1) {
for(k=0; k<j; k=k+1) {
draw(shift(5*i-5+(i-2)*(i-1)*1,0)*shift(2(j-k)+k, k*r)*p);
}}}[/asy]
The length of the side of the smallest triangles is $1$. A triangle is called a delta if its vertex is at the top; for example, there are $10$ deltas in $T_{3}$. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in $T_{20}$?
2017 CHMMC (Fall), 2
Adam the spider is sitting at the bottom left of a 4 × 4 coordinate grid, where adjacent parallel grid lines are each separated by one unit. He wants to crawl to the top right corner of the square, and starts off with 9 “crumb’s” worth of energy. Adam only walks in one-unit segments along the grid lines, and cannot walk off of the grid.
Walking one unit costs him one crumb’s worth of energy, and Adam cannot move anymore once he runs out of energy. Also, Adam stops moving once he reaches the top right corner.
There is also a single crumb on the grid located one unit to the right and one unit up from Adam’s starting position. If he goes to this point and eats the crumb, he will gain one crumb’s worth of energy.
How many paths can Adam take to get to the upper right corner of the grid? Note that Adam does not care if he has extra energy left over once he arrives at his destination.
2020 Thailand TST, 4
For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.
1989 Federal Competition For Advanced Students, P2, 5
Find all real solutions of the system:
$ x^2\plus{}2yz\equal{}x,$
$ y^2\plus{}2zx\equal{}y,$
$ z^2\plus{}2xy\equal{}z.$
2002 National Olympiad First Round, 10
Which of the following does not divide the number of ordered pairs $(x,y)$ of integers satisfying the equation $x^3 - 13y^3 = 1453$?
$
\textbf{a)}\ 2
\qquad\textbf{b)}\ 3
\qquad\textbf{c)}\ 5
\qquad\textbf{d)}\ 7
\qquad\textbf{e)}\ \text{None of above}
$
1989 AMC 12/AHSME, 9
Mr. and Mrs. Zeta want to name their baby Zeta so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated. How many such monograms are possible?
$\text{(A)} \ 276 \qquad \text{(B)} \ 300 \qquad \text{(C)} \ 552 \qquad \text{(D)} \ 600 \qquad \text{(E)} \ 15600$
Math Hour Olympiad, Grades 8-10, 2014.4
Hermione and Ron play a game that starts with $129$ hats arranged in a circle. They take turns magically transforming the hats into animals. On each turn, a player picks a hat and chooses whether to change it into a badger or into a raven.
A player loses if after his or her turn there are two animals of the same species right next to
each other. Hermione goes first. Who loses?
2003 Paraguay Mathematical Olympiad, 4
Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]