Found problems: 85335
2002 AMC 10, 8
How many ordered triples of positive integers $(x,y,z)$ satisfy $(x^y)^z=64$?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2019 IMO Shortlist, G4
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
2022 Putnam, A4
Suppose that $X_1, X_2, \ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum_{i=1}^kX_i/2^i,$ where $k$ is the least positive integer such that $X_k<X_{k+1},$ or $k=\infty$ if there is no such integer. Find the expected value of $S.$
2012 ELMO Shortlist, 6
In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear.
[i]Ray Li.[/i]
2011 ELMO Shortlist, 2
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
1952 Moscow Mathematical Olympiad, 212
Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.
2024 Ukraine National Mathematical Olympiad, Problem 2
For some positive integer $n$, consider the board $n\times n$. On this board you can put any rectangles with sides along the sides of the grid. What is the smallest number of such rectangles that must be placed so that all the cells of the board are covered by distinct numbers of rectangles (possibly $0$)? The rectangles are allowed to have the same sizes.
[i]Proposed by Anton Trygub[/i]
2021 Science ON grade VI, 2
Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of $36^o$?
[i] (Adapted from Archimedes 2011, Traian Preda)[/i]
2018 Singapore Senior Math Olympiad, 1
You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?
2022 Purple Comet Problems, 18
In $\vartriangle ABC$, let $D$ be on $BC$ such that $\overline{AD} \perp \overline{BC}$. Suppose also that $\tan B = 4 \sin C$, $AB^2 +CD^2 = 17$, and $AC^2 + BC^2 = 21$. Find the measure of $\angle C$ in degrees between $0^o$ and $180^o$ .
2022 Moldova EGMO TST, 11
Let there be a trapezoid $ABCD$ with bases $AD$ and $BC$. Points $M$ and $P$ are on sides $AB$ and $CD$ such that $CM$ and $BP$ intersect in $N$ and the pentagon $AMNPD$ is cyclic. Prove that the triangle $ADN$ is isosceles.
2012 Brazil Team Selection Test, 4
Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$
2016 Online Math Open Problems, 17
A set $S \subseteq \mathbb{N}$ satisfies the following conditions:
(a) If $x, y \in S$ (not necessarily distinct), then $x + y \in S$.
(b) If $x$ is an integer and $2x \in S$, then $x \in S$.
Find the number of pairs of integers $(a, b)$ with $1 \le a, b\le 50$ such that if $a, b \in S$ then $S = \mathbb{N}.$
[i] Proposed by Yang Liu [/i]
1994 IMO Shortlist, 1
Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$
2015 USAMTS Problems, 2
A net for a polyhedron is cut along an edge to give two [b]pieces[/b]. For example, we may cut a cube net along the red edge to form two pieces as shown.
[asy]
size(5.5cm);
draw((1,0)--(1,4)--(2,4)--(2,0)--cycle);
draw((1,1)--(2,1));
draw((1,2)--(2,2));
draw((1,3)--(2,3));
draw((0,1)--(3,1)--(3,2)--(0,2)--cycle);
draw((2,1)--(2,2),red+linewidth(1.5));
draw((3.5,2)--(5,2));
filldraw((4.25,2.2)--(5,2)--(4.25,1.8)--cycle,black);
draw((6,1.5)--(10,1.5)--(10,2.5)--(6,2.5)--cycle);
draw((7,1.5)--(7,2.5));
draw((8,1.5)--(8,2.5));
draw((9,1.5)--(9,2.5));
draw((7,2.5)--(7,3.5)--(8,3.5)--(8,2.5)--cycle);
draw((11,1.5)--(11,2.5)--(12,2.5)--(12,1.5)--cycle);
[/asy]
Are there two distinct polyhedra for which this process may result in the same two pairs of pieces? If you think the answer is no, prove that no pair of polyhedra can result in the same two pairs of pieces. If you think the answer is yes, provide an example; a clear example will suffice as a proof.
2007 Korea Junior Math Olympiad, 2
If $n$ is a positive integer and $a, b$ are relatively prime positive integers, calculate $(a + b,a^n + b^n)$.
2009 Germany Team Selection Test, 3
The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.
2023 LMT Spring, 10
Positive integers $a$, $b$, and $c$ satisfy $a^2 +b^2 = c^3 -1$ where $c \le 40$. Find the sum of all distinct possible values of $c$.
1970 IMO Longlists, 59
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
2000 Poland - Second Round, 4
Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.
1997 Moscow Mathematical Olympiad, 5
Let $1+x+x^2+...+x^{n-1}=F(x)G(x)$, where $n>1$ and where $F$ and $G$ are polynomials whose coefficients are zeroes and units. Prove that one of the polynomials $F$ and $G$ can be represented in the form $(1+x+x^2+...x^{k-1})T(x),$ where $k>1$ and $T$ is a polynomial whose coefficients are zeroes and units.
1995 Belarus Team Selection Test, 1
There is a 100 x100 square table, a real number being written in each cell.$A$ and $B$ play the following game. They choose, turn by turn, some row of the table (if it has not been chosen before). When $A$ and $B$ have $50$ rows chosen each, they sum the numbers in the corresponding cells of the chosen rows, and then sum the squares of all $100$ obtained numbers and compare the results. $A$ player who has the greater result wins. Player $A$ begins. Show that $A$ can avoid a defeat.
2014 JHMMC 7 Contest, 9
Let $n!=n\cdot (n-1)\cdot (n-2)\cdot \ldots \cdot 2\cdot 1$.For example, $5! = 5\cdot 4\cdot 3 \cdot 2\cdot 1 = 120.$ Compute $\frac{(6!)^2}{5!\cdot 7!}$.
2005 Iran Team Selection Test, 2
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
2016 IFYM, Sozopol, 1
Find all functions $f: \mathbb{R}^+\rightarrow \mathbb{R}^+$ with the following property: $a,b,$ and $c$ are lengths of sides of a triangle, if and only if $f(a),f(b),$ and $f(c)$ are lengths of sides of a triangle.