Found problems: 85335
Let $ABC$ be a triangle whose angle at $A$ is right, and let $D$ be the foot of the altitude from $A$. A variable point $M$ traces the interior of the minor arc $AB$ of the circle $ABC$. The internal bisector of the angle $DAM$ crosses $CM$ at $N$. The line through $N$ and perpendicular to $CM$ crosses the line $AD$ at $P$. Determine the locus of the point where the line $BN$ crosses the line $CP$.
[i]* * *[/i]
An ant is crawling along the coordinate plane. Each move, it moves one unit up, down, left, or right with equal probability. If it starts at $(0,0)$, what is the probability that it will be at either $(2,1)$ or $(1,2)$ after $6$ moves?
[i]2020 CCA Math Bonanza Individual Round #1[/i]
Find all possible finite sequences $\{n_0, n_1, n_2, \ldots, n_k \}$ of integers such that for each $i, i$ appears in the sequence $n_i$ times $(0 \leq i \leq k).$
Let $ABCD$ be a quadrilateral with all the internal angles $< \pi$. Squares are drawn on each side as shown in the picture below. Let $\Delta_1 , \Delta_2 , \Delta_3 , \Delta_4$ denote the areas of the shaded triangles as shown. Prove that \[\Delta_1 - \Delta_2 + \Delta_3 - \Delta_4 = 0.\]
[asy]
//made from sweat and hardwork by SatisfiedMagma
import olympiad;
import geometry;
size(250);
pair A = (-3,0);
pair B = (0,2);
pair C = (5.88,0.44);
pair D = (0.96, -1.86);
pair H = B + rotate(90)*(C-B);
pair G = C + rotate(270)*(B-C);
pair J = C + rotate(90)*(D-C);
pair I = D + rotate(270)*(C-D);
pair L = D + rotate(90)*(A-D);
pair K = A + rotate(270)*(D-A);
pair F = A + rotate(90)*(B-A);
pair E = B + rotate(270)*(A-B);
draw(B--H--G--C--B, blue);
draw(C--J--I--D--C, red);
draw(B--E--F--A--B, orange);
draw(D--L--K--A--D, magenta);
draw(L--I, fuchsia); draw(J--G, fuchsia); draw(E--H, fuchsia); draw(F--K, fuchsia);
pen lightFuchsia = deepgreen + 0.5*white;
fill(D--L--I--cycle, lightFuchsia);
fill(A--K--F--cycle, lightFuchsia);
fill(E--B--H--cycle, lightFuchsia);
fill(C--J--G--cycle, lightFuchsia);
label("$\triangle_2$", (E+B+H)/3);
label("$\triangle_4$", (D+L+I)/3);
label("$\triangle_3$", (C+G+J)/3);
label("$\triangle_1$", (A+F+K)/3);
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, S);
dot("$D$", D, N);
dot("$H$", H, dir(H));
dot("$G$", G, dir(G));
dot("$J$", J, dir(J));
dot("$I$", I, dir(I));
dot("$L$", L, dir(L));
dot("$K$", K, dir(K));
dot("$F$", F, dir(F));
dot("$E$", E, dir(E));
[/asy]
Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of
some $ n$ consecutive integers is a perfect square?
Lets say that a positive integer is $good$ if its equal to the the subtraction of two positive integer cubes. For example: $7$ is a $good$ prime because $2^3-1^3=7$.
Determine how much the last digit of a $good$ prime may be worth. Give all the possibilities.
Let $n$, $(n \geq 3)$ be a positive integer and the set $A$={$1,2,...,n$}. All the elements of $A$ are randomly arranged in a sequence $(a_1,a_2,...,a_n)$. The pair $(a_i,a_j)$ forms an $inversion$ if $1 \leq i \leq j \leq n$ and $a_i > a_j$. In how many different ways all the elements of the set $A$ can be arranged in a sequence that contains exactly $3$ inversions?
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that:
\[a^p+b^p=p^c.\]
Consider an acute triangle $ABC$ of area $S$. Let $CD \perp AB$ ($D \in AB$), $DM \perp AC$ ($M \in AC$) and $DN \perp BC$ ($N \in BC$). Denote by $H_1$ and $H_2$ the orthocentres of the triangles $MNC$, respectively $MND$. Find the area of the quadrilateral $AH_1BH_2$ in terms of $S$.
Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.
Three circles are constructed for the triangle $ABC $: the circle ${{w} _ {A}} $ passes through the vertices $B $ and $C $ and intersects the sides $AB $ and $ AC $ at points ${{A} _ {1}} $ and ${{A} _ {2}} $ respectively, the circle ${{w} _ {B}} $ passes through the vertices $A $ and $C $ and intersects the sides $BA $ and $BC $ at the points ${{B} _ {1}} $ and ${{B} _ {2}} $, ${{w} _ {C}} $ passes through the vertices $A $ and $B $ and intersects the sides $CA $ and $CB $ at the points ${{C} _ {1}} $ and ${{C} _ {2}} $. Let ${{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '$, ${{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '$ ta ${ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '$ is Prove that the perpendiculars, which are omitted from the points ${A} ', \, \, {B}', \, \, {C} '$ to the lines $BC $, $CA $ and $AB $ respectively intersect at one point.
(Rudenko Alexander)
Let $x_1,x_2,\dots ,x_6$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5+x_6=1$, and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$. Find $p+q$.
One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle
Given that
\[\frac{a-b}{c-d}=2\quad\text{and}\quad\frac{a-c}{b-d}=3\]
for certain real numbers $a,b,c,d$, determine the value of
\[\frac{a-d}{b-c}.\]
Because J’s cheetahs are everywhere, they are now running away. A particularly unintelligent one starts to run in a $720$ mile loop at $80$ miles per hour. J immediately starts to chase after it, starting at the same point, at $10$ miles per hour at $12:00$ PM, but realizes one hour later that it would be more wise to turn around and run in the opposite direction in the loop, so he does this. Assuming both maintain a constant speed, at what time do J and the cheetah collide? Round to the nearest minute, and be sure to include AM or PM.
Two pirates divide the loot, consisting of two bags of coins and a diamond, according to the following rules. First the first pirate takes take a few coins from any bag and transfer them from this bag in the other the same number of coins. Then the second pirate does the same (choosing the bag from which he takes the coins at his discretion) and etc. until you can take coins according to these rules. The pirate who takes the coins last gets the diamond. Who will get the diamond if is each of the pirates trying to get it? Give your answer depending on the initial number of coins in the bags.
A shopper plans to purchase an item that has a listed price greater than $ \$100$ and can use any one of the three coupons. Coupon A gives $ 15\%$ off the listed price, Coupon B gives $ \$30$ the listed price, and Coupon C gives $ 25\%$ off the amount by which the listed price exceeds $ \$100$.
Let $ x$ and $ y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $ y\minus{}x$?
$ \textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 80\qquad\textbf{(E)}\ 100$
Let $ G $ be a countably infinite, $ d $ -regular, connected, vertex-transitive graph. Show that there is a complete pairing in $ G $.
Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( 0 < f'(t) \leq 1 \) for all \( t \in [0, 1] \). Show that:
\[
\left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt.
\]
For composite number $n$, let $f(n)$ denote the sum of the least three divisors of $n$, and $g(n)$ the sum of the greatest two divisors of $n$. Find all composite numbers $n$, such that $g(n)=(f(n))^m$ ($m\in N^*$).
A positive integer $a$ is said to [i]reduce[/i] to a positive integer $b$ if when dividing $a$ by its units digits the result is $b$. For example, 2015 reduces to $\frac{2015}{5} = 403$.
Find all the positive integers that become 1 after some amount of reductions. For example, 12 is one such number because 12 reduces to 6 and 6 reduces to 1.
Solve the following equation in $\mathbb{Z}$:
\[3^{2a + 1}b^2 + 1 = 2^c\]
Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$
[i]Proposed by Nikola Velov[/i]
Evaluate: $\lim_{n\rightarrow\infty}\sum_{k=n^2}^{(n+1)^2} \dfrac{1}{\sqrt{k}}$
Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.