Found problems: 85335
2022 MIG, 16
An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters?
$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$
2009 Putnam, B5
Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that
\[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\]
Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$
2013 AMC 12/AHSME, 9
In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?
[asy]
size(180);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
real r=5/7;
pair A=(10,sqrt(28^2-100)),B=origin,C=(20,0),D=(A.x*r,A.y*r);
pair bottom=(C.x+(D.x-A.x),C.y+(D.y-A.y));
pair E=extension(D,bottom,B,C);
pair top=(E.x+D.x,E.y+D.y);
pair F=extension(E,top,A,C);
draw(A--B--C--cycle^^D--E--F);
dot(A^^B^^C^^D^^E^^F);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,W);
label("$E$",E,S);
label("$F$",F,dir(0));
[/asy]
$\textbf{(A) }48\qquad
\textbf{(B) }52\qquad
\textbf{(C) }56\qquad
\textbf{(D) }60\qquad
\textbf{(E) }72\qquad$
2013 AMC 10, 25
All diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?
$\textbf{(A)} \ 49 \qquad \textbf{(B)} \ 65 \qquad \textbf{(C)} \ 70 \qquad \textbf{(D)} \ 96 \qquad \textbf{(E)} \ 128$
2017-2018 SDPC, 5
Given positive real numbers $a,b,c$ such that $abc=1$, find the maximum possible value of $$\frac{1}{(4a+4b+c)^3}+\frac{1}{(4b+4c+a)^3}+\frac{1}{(4c+4a+b)^3}.$$
1978 IMO Shortlist, 16
Determine all the triples $(a, b, c)$ of positive real numbers such that the system
\[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\]
is compatible in the set of real numbers, and then find all its real solutions.
2011 JHMT, 5
Let $ABCD$ be a unit square. Point $E$ is on $BC$, point $F$ is on $DC$, $\vartriangle AEF$ is equilateral, and $GHIJ$ is a square in $\vartriangle AEF$ such that $GH$ is on $EF$. Compute the area of square $GHIJ$.
2022 Rioplatense Mathematical Olympiad, 2
Four teams $A$, $B$, $C$ and $D$ play a football tournament in which each team plays exactly two times against each of the remaining three teams (there are $12$ matches). In each matchif it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $3$ points and the loser gets $0$ points.
At the end of the tournament the teams $A$, $B$ and $C$ have $8$ points each. Determine all possible points of team $D$.
2014 Contests, 3
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
1952 Putnam, A3
Develop necessary and sufficient conditions which ensure that $r_1, r_2, r_3$ and $r_1^2, r_2^2, r_3^2$ are simultaneously roots of the equation $x^3 + ax^2 + bx + c = 0.$
1981 IMO Shortlist, 10
Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).
1940 Putnam, B4
Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface
$$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$
is the sphere
$$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$
Estonia Open Junior - geometry, 2003.1.2
Circles with centres $O_1$ and $O_2$ intersect in two points, let one of which be $A$. The common tangent of these circles touches them respectively in points $P$ and $Q$. It is known that points $O_1, A$ and $Q$ are on a common straight line and points $O_2, A$ and $P$ are on a common straight line. Prove that the radii of the circles are equal.
2018 Junior Balkan Team Selection Tests - Romania, 3
Let $D$ be a unique point on segment $BC$, in $ABC$. If $AD^2 = BD \cdot CD$, show that $AB + AC = \sqrt{2}BC$.
2016 JBMO Shortlist, 5
Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$ lies on the circumcircle of ${ABC}$. Reflect $O$ across ${X}$ to obtain ${O'}$, and let the lines ${XH}$and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right],\left[ XC \right]$ and $\left[ BC \right]$, respectively. Prove that the points $K,L,M$ and ${N}$ are concyclic.
2006 IberoAmerican, 1
Find all pairs $(a,\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$
2019 Tuymaada Olympiad, 8
Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.
2018 IFYM, Sozopol, 1
Find all prime numbers $p$ and all positive integers $n$, such that
$n^8 - n^2 = p^5 + p^2$
2018 Indonesia Juniors, day 1
The problems are really difficult to find online, so here are the problems.
P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form.
a) Determine the smallest possible value of $k$.
b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$.
c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.
I added this because my translation is a bit weird.
[hide=Indonesian Version] Diketahui dua bilangan bulat positif $m$ dan $n$ dengan $10n - 9m = 7$ dan $m \leq 2018$. Bilangan $k = 20 - \frac{18m}{n}$ merupakan suatu pecahan sederhana.
a) Tentukan bilangan $k$ terkecil yang mungkin.
b) Jika penyebut bilangan $k$ terkecil tersebut adalah $N$, tentukan semua faktor positif dari $N$.
c) Pada pengambilan satu faktor dari faktor-faktor positif $N$ di atas, tentukan peluang terambilnya satu faktor kelipatan 4.[/hide]
P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs.
[hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description:
For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with $X$ or $Y$ depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their $X$ or $Y$ coordinates only (since their other coordinates are definitely 0).
Graph (1) is the function $f$, who is a quadratic function with -2 and 4 as its $X$-intercepts and 4 as its $Y$-intercept. You also put $f$ right besides the curve you have, preferably just on the right-up direction of said curve.
Graph (2) is the function $g$, which is piecewise. For $x \geq 0$, $g(x) = \frac{1}{2}x - 2$, whereas for $x < 0$, $g(x) = - x - 2$. You also put $g$ right besides the curve you have, on the lower right of the line, on approximately $x = 2$.[/hide]
Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$.
a) Draw the graph of the function $g \circ f$.
b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$.
P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$.
P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$.
P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.
2016 Saudi Arabia Pre-TST, 1.2
Let $a, b, c$ be positive numbers such that $a^2+b^2+c^2+abc = 4$.
Prove that $$\frac{a + b}{c} +\frac{b + c}{a} +\frac{c + a}{b} \ge a + b + c + \frac{1}{a} + \frac{1}{b} +\frac{1}{c}$$
2002 Tournament Of Towns, 4
There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.
1980 Bulgaria National Olympiad, Problem 1
Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.
1940 Moscow Mathematical Olympiad, 061
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
2018 Pan-African Shortlist, G4
Let $ABC$ be a triangle and $\Gamma$ be the circle with diameter $[AB]$. The bisectors of $\angle BAC$ and $\angle ABC$ cut the circle $\Gamma$ again at $D$ and $E$, respectively. The incicrcle of the triangle $ABC$ cuts the lines $BC$ and $AC$ in $F$ and $G$, respectively. Show that the points $D, E, F$ and $G$ lie on the same line.
1955 Miklós Schweitzer, 6
[b]6.[/b] For a prime factorisation of a positive integer $N$ let us call the exponent of a prime $p$ the integer $k$ for which $p^{k} \mid N$ but $p^{k+1} \nmid N$; let, further, the power $p^{k}$ be called the "contribution" of $p$ to $N$. Show that for any positive integer $n$ and for any primes $p$ and $q$ the contibution of $p$ to $n!$ is greater than the contribution of $q$ if and only if the exponent of $p$ is greater than that of $q$.