Found problems: 85335
2001 National High School Mathematics League, 3
An $m\times n(m,n\in \mathbb{Z}_+)$ rectangle is divided into some smaller squares. All sides of each square are parallel to the sides of the rectangle, and the length of each side is an integer. Determine the minimum value of the sum of the lengths of sides of these squares.
2024 Chile TST Ibero., 5
Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.
1979 Bulgaria National Olympiad, Problem 3
Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.
2007 Germany Team Selection Test, 3
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.
[i]Proposed by J.P. Grossman, Canada[/i]
2022 China Team Selection Test, 3
Find all functions $f: \mathbb R \to \mathbb R$ such that for any $x,y \in \mathbb R$, the multiset $\{(f(xf(y)+1),f(yf(x)-1)\}$ is identical to the multiset $\{xf(f(y))+1,yf(f(x))-1\}$.
[i]Note:[/i] The multiset $\{a,b\}$ is identical to the multiset $\{c,d\}$ if and only if $a=c,b=d$ or $a=d,b=c$.
2023 ISL, C3
Let $n$ be a positive integer. A [i]Japanese triangle[/i] consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\dots$, $n$, the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured red. A [i]ninja path[/i] in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles.
[asy]
// credit to vEnhance for the diagram (which was better than my original asy):
size(4cm);
pair X = dir(240); pair Y = dir(0);
path c = scale(0.5)*unitcircle;
int[] t = {0,0,2,2,3,0};
for (int i=0; i<=5; ++i) {
for (int j=0; j<=i; ++j) {
filldraw(shift(i*X+j*Y)*c, (t[i]==j) ? lightred : white);
draw(shift(i*X+j*Y)*c);
}
}
draw((0,0)--(X+Y)--(2*X+Y)--(3*X+2*Y)--(4*X+2*Y)--(5*X+2*Y),linewidth(1.5));
path q = (3,-3sqrt(3))--(-3,-3sqrt(3));
draw(q,Arrows(TeXHead, 1));
label("$n = 6$", q, S);
label("$n = 6$", q, S);
[/asy]
In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.
2019 Hong Kong TST, 6
If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of
\[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \]
1997 Brazil Team Selection Test, Problem 5
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.
(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.
(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
2021 AMC 12/AHSME Fall, 17
How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?
$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$
2012 Mediterranean Mathematics Olympiad, 1
For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and
\[ \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
(Proposed by Gerhard Woeginger, Austria)
2006 Sharygin Geometry Olympiad, 12
In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.
2008 Alexandru Myller, 1
How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have?
[i]Mihail Bălună[/i]
1983 National High School Mathematics League, 2
Function $f(x)$ is defined on $[0,1]$, $f(0)=f(1)$. For any $x_1,x_2\in [0,1], |f(x_1)-f(x_2)|<|x_1-x_2|(x_1\neq x_2)$. Prove that $|f(x_1)-f(x_2)|<\frac{1}{2}$.
1997 Taiwan National Olympiad, 4
Let $k=2^{2^{n}}+1$ for some $n\in\mathbb{N}$. Show that $k$ is prime iff $k|3^{\frac{k-1}{2}}+1$.
2009 CHKMO, 1
Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$.
(a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$.
(b) Show that $ a_{2008} \neq 0$
2021 LMT Fall, 7
Find the number of ways to tile a $12 \times 3$ board with $1 \times 4$ and $2 \times 2$ tiles with no overlap or uncovered space.
2010 Iran MO (2nd Round), 5
In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.
CNCM Online Round 1, 7
Three cats--TheInnocentKitten, TheNeutralKitten, and TheGuiltyKitten labelled $P_1, P_2,$ and $P_3$ respectively with $P_{n+3} = P_{n}$--are playing a game with three rounds as follows:
[list=1]
[*] Each round has three turns. For round $r \in \{1,2,3\}$ and turn $t \in \{1,2,3\}$ in that round, player $P_{t+1-r}$ picks a non-negative integer. The turns in each round occur in increasing order of $t$, and the rounds occur in increasing order of $r$.
$\newline
\newline$
[*] [b]Motivations:[/b] Every player focuses primarily on maximizing the sum of their own choices and secondarily on minimizing the total of the other players’ sums. TheNeutralKitten and TheGuiltyKitten have the additional tertiary priority of minimizing TheInnocentKitten’s sum.
$\newline
\newline$
[*] For round $2$, player $P_{2}$ has no choice but to pick the number equal to what player $P_{1}$ chose in round $1$. Likewise, for round $3$, player $P_{3}$ must pick the number equal to what player $P_{2}$ chose in round $2$.
$\newline
\newline$
[*] If not all three players choose their numbers such that the values they chose in rounds 1,2,3 form an arithmetic progression in that order by the end of the game, all players' sums are set to $-1$ regardless of what they have chosen.
$\newline
\newline$
[*] If the sum of the choices in any given round is greater than $100$, all choices that round are set to $0$ at the end of that round. That is, rules $2$, $3$, and $4$ act as if each player chose $0$ that round.
$\newline
\newline$
[*] All players play optimally as per their motivations. Furthermore, all players know that all other players will play optimally (and so on.)
[/list]
Let $A$ and $B$ be TheInnocentKitten's sum and TheGuiltyKitten's sum respectively. Compute $1000A + B$ when all players play optimally.
Proposed by Harry Chen (Extile)
1987 Bundeswettbewerb Mathematik, 1
Find all non-negative integer solutions of the equation
\[2^x + 3^y = z^2 .\]
2014 Bosnia and Herzegovina Junior BMO TST, 1
Let $x$, $y$ and $z$ be nonnegative integers. Find all numbers in form $\overline{13xy45z}$ divisible with $792$, where $x$, $y$ and $z$ are digits.
2014 CHMMC (Fall), 9
There is a long-standing conjecture that there is no number with $2n + 1$ instances in Pascal’s triangle for $n \ge 2$. Assuming this is true, for how many $n \le 100, 000$ are there exactly $3$ instances of $n$ in Pascal’s triangle?
2024 USA TSTST, 4
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$.
[i]Merlijn Staps[/i]
2019 IMEO, 5
Find all pairs of positive integers $(s, t)$, so that for any two different positive integers $a$ and $b$ there exists some positive integer $n$, for which $$a^s + b^t | a^n + b^{n+1}.$$
[i]Proposed by Oleksii Masalitin (Ukraine)[/i]
2000 Argentina National Olympiad, 1
The natural numbers are written in succession, forming a sequence of digits$$12345678910111213141516171819202122232425262728293031\ldots$$Determine how many digits the natural number has that contributes to this sequence with the digit in position $10^{2000}$.
Clarification: The natural number that contributes to the sequence with the digit in position $10$ has $2$ digits, because it is $10$; The natural number that contributes to the sequence with the digit at position $10^2$ has $2$ digits, because it is $55$.
1990 Iran MO (2nd round), 3
[b](a)[/b] For every positive integer $n$ prove that
\[1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2\]
[b](b)[/b] Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that
\[\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1\]