Found problems: 85335
2019 Ecuador NMO (OMEC), 3
For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$
2010 IMO Shortlist, 5
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]
[i]Proposed by Thomas Huber, Switzerland[/i]
2017 ASDAN Math Tournament, 2
Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$?
2022 Math Prize for Girls Olympiad, 4
Let $n > 1$ be an integer. Let $A$ denote the set of divisors of $n$ that are less than $\sqrt n$. Let $B$ denote the set of divisors of $n$ that are greater than $\sqrt n$. Prove that there exists a bijective function $f \colon A \to B$ such that $a$ divides $f(a)$ for all $a \in A$.
(We say $f$ is [i]bijective[/i] if for every $b \in B$ there exists a unique $a \in A$ with $f(a) = b$.)
2019 Slovenia Team Selection Test, 5
Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$.
Prove that, $PQ$ bisects $BC$.
2025 Azerbaijan Senior NMO, 1
Alice creates a sequence: For the first $2025$ terms of this sequence, she writes a random permutation of $\{1;2;3;...;2025\}$. To define the following terms, she does the following: She takes the last $2025$ terms of the sequence, and takes its median. How many values could this sequence's $3000$'th term could get?
(Note: To find the median of $2025$ numbers, you write them in an increasing order,and take the number in the middle)
2000 Belarus Team Selection Test, 6.2
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$.
Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.
2007 Tournament Of Towns, 6
Let $a_0$ be an irrational number such that $0 < a_0 < \frac 12$ . Defi ne $a_n = \min \{2a_{n-1},1 - 2a_{n-1}\}$ for $n \geq 1$.
[list][b](a)[/b] Prove that $a_n < \frac{3}{16}$ for some $n$.
[b](b)[/b] Can it happen that $a_n > \frac{7}{40}$ for all $n$?[/list]
2010 Princeton University Math Competition, 6
A regular pentagon is drawn in the plane, along with all its diagonals. All its sides and diagonals are extended infinitely in both directions, dividing the plane into regions, some of which are unbounded. An ant starts in the center of the pentagon, and every second, the ant randomly chooses one of the edges of the region it's in, with an equal probability of choosing each edge, and crosses that edge into another region. If the ant enters an unbounded region, it explodes. After first leaving the central region of the pentagon, let $x$ be the expected number of times the ant re-enters the central region before it explodes. Find the closest integer to $100x$.
1978 AMC 12/AHSME, 1
If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals
$\textbf{(A) }-1\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$
2006 Korea National Olympiad, 3
For three positive integers $a,b$ and $c,$ if $\text{gcd}(a,b,c)=1$ and $a^2+b^2+c^2=2(ab+bc+ca),$ prove that all of $a,b,c$ is perfect square.
2023/2024 Tournament of Towns, 2
2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide?
Alexandr Yuran
2018 CHMMC (Fall), 8
Find the largest positive integer $n$ that cannot be written as $n = 20a + 28b + 35c$ for nonnegative integers $a, b$, and $c$.
2002 Vietnam National Olympiad, 3
Let be given two positive integers $ m$, $ n$ with $ m < 2001$, $ n < 2002$. Let distinct real numbers be written in the cells of a $ 2001 \times 2002$ board (with $ 2001$ rows and $ 2002$ columns). A cell of the board is called [i]bad[/i] if the corresponding number is smaller than at least $ m$ numbers in the same column and at least $ n$ numbers in the same row. Let $ s$ denote the total number of [i]bad[/i] cells. Find the least possible value of $ s$.
2006 AMC 8, 25
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers?
[asy]path card=((0,0)--(0,3)--(2,3)--(2,0)--cycle);
draw(card, linewidth(1));
draw(shift(2.5,0)*card, linewidth(1));
draw(shift(5,0)*card, linewidth(1));
label("$44$", (1,1.5));
label("$59$", shift(2.5,0)*(1,1.5));
label("$38$", shift(5,0)*(1,1.5));[/asy]
$ \textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 14 \qquad
\textbf{(C)}\ 15 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17$
2016 Online Math Open Problems, 30
Let $P_1(x),P_2(x),\ldots,P_n(x)$ be monic, non-constant polynomials with integer coefficients and let $Q(x)$ be a polynomial with integer coefficients such that \[x^{2^{2016}}+x+1=P_1(x)P_2(x)\ldots P_n(x)+2Q(x).\] Suppose that the maximum possible value of $2016n$ can be written in the form $2^{b_1}+2^{b_2}+\cdots+2^{b_k}$ for nonnegative integers $b_1<$ $b_2<$ $\cdots<$ $b_k$. Find the value of $b_1+b_2+\cdots+b_k$.
[i]Proposed by Michael Ren[/i]
1987 Bundeswettbewerb Mathematik, 1
Find all non-negative integer solutions of the equation
\[2^x + 3^y = z^2 .\]
1993 ITAMO, 5
Prove the following inequality for any positive real numbers a,b,c not exceeding 1
$a^2b+b^2c+c^2a+1\ge a^2+b^2+c^2$
1994 Irish Math Olympiad, 4
Suppose that $ \omega, a,b,c$ are distinct real numbers for which there exist real numbers $ x,y,z$ that satisfy the following equations:
$ x\plus{}y\plus{}z\equal{}1,$
$ a^2 x\plus{}b^2 y \plus{}c^2 z\equal{}\omega ^2,$
$ a^3 x\plus{}b^3 y \plus{}c^3 z\equal{}\omega ^3,$
$ a^4 x\plus{}b^4 y \plus{}c^4 z\equal{}\omega ^4.$
Express $ \omega$ in terms of $ a,b,c$.
1997 IMO Shortlist, 23
Let $ ABCD$ be a convex quadrilateral. The diagonals $ AC$ and $ BD$ intersect at $ K$. Show that $ ABCD$ is cyclic if and only if $ AK \sin A \plus{} CK \sin C \equal{} BK \sin B \plus{} DK \sin D$.
2013 Tuymaada Olympiad, 6
Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$ coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root.
[i]K. Kokhas & F. Petrov[/i]
2002 Tuymaada Olympiad, 8
The circle with the center of $ O $ touches the sides of the angle $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.
1979 Dutch Mathematical Olympiad, 4
Given is the non-equilateral triangle $A_1A_2A_3$. $B_{ij}$ is the symmetric of $A_i$ wrt the inner bisector of $\angle A_j$. Prove that lines $B_{12}B_{21}$, $B_{13}B_{31}$ and $B_{23}B_{32}$ are parallel.
1970 Vietnam National Olympiad, 1
Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.
1996 AMC 8, 2
Jose, Thuy, and Kareem each start with the number $10$. Jose subtracts $1$ from the number $10$, doubles his answer, and then adds $2$. Thuy doubles the number $10$, subtracts $1$ from her answer, and then adds $2$. Kareem subtracts $1$ from the number $10$, adds $2$ to his number, and then doubles the result. Who gets the largest final answer?
$\text{(A)}\ \text{Jose} \qquad \text{(B)}\ \text{Thuy} \qquad \text{(C)}\ \text{Kareem} \qquad \text{(D)}\ \text{Jose and Thuy} \qquad \text{(E)}\ \text{Thuy and Kareem}$