Found problems: 85335
2005 Cono Sur Olympiad, 2
Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively.
(a) Show that $R$, $Q$, $W$, $S$ are collinear.
(b) Show that $MP=RS-QW$.
1910 Eotvos Mathematical Competition, 1
If $a, b, c$ are real numbers such that $$a^2 + b^2 + c^2 = 1$$ prove the inequalities $$- \frac12 \le ab + bc + ca \le 1$$
1988 AMC 12/AHSME, 5
If $b$ and $c$ are constants and \[(x + 2)(x + b) = x^2 + cx + 6,\] then $c$ is
$ \textbf{(A)}\ -5\qquad\textbf{(B)}\ -3\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 5 $
1996 AIME Problems, 7
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
2022 AMC 10, 2
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$?
[asy]
import olympiad;
size(180);
real r = 3, s = 5, t = sqrt(r*r+s*s);
defaultpen(linewidth(0.6) + fontsize(10));
pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0);
draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D));
label("$A$",A,SW);
label("$B$", B, NW);
label("$C$",C,NE);
label("$D$",D,SE);
label("$P$",P,S);
[/asy]
$\textbf{(A) }3\sqrt 5 \qquad
\textbf{(B) }10 \qquad
\textbf{(C) }6\sqrt 5 \qquad
\textbf{(D) }20\qquad
\textbf{(E) }25$
2020 Azerbaijan IZHO TST, 6
Define a sequence ${{a_n}}_{n\ge1}$ such that $a_1=1$ , $a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $gcd(m,a_n)\neq{1}$. Show that all positive integers occur in the sequence.
1998 AMC 8, 9
For a sale, a store owner reduces the price of a $10$ dollar scarf by $20\%$. Later the price is lowered again, this time by one-half the reduced price. The price is now
$ \text{(A)}\ 2.00\text{ dollars}\qquad\text{(B)}\ 3.75\text{ dollars}\qquad\text{(C)}\ 4.00\text{ dollars}\qquad\text{(D)}\ 4.90\text{ dollars}\qquad\text{(E)}\ 6.40\text{ dollars} $
2023 India Regional Mathematical Olympiad, 1
Let $\mathbb{N}$ be the set of all positive integers and $S=\left\{(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2\right\}$. Find the largest positive integer $m$ such that $m$ divides abcd for all $(a, b, c, d) \in S$.
2006 All-Russian Olympiad Regional Round, 10.2
We call a coloring of an $8\times 8$ board in three colors good if in any corner of five cells contains cells of all three colors. (A five-square corner is a shape made from a $3 \times 3$ square by cutting square $ 2\times 2$.) Prove that the number of good colorings is not less than than $68$.
2010 Rioplatense Mathematical Olympiad, Level 3, 1
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$.
[list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest?
(b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]
2001 AMC 12/AHSME, 7
A charity sells 140 benefit tickets for a total of $ \$2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?
$ \textbf{(A)} \ \$782 \qquad \textbf{(B)} \ \$986 \qquad \textbf{(C)} \ \$1158 \qquad \textbf{(D)} \ \$1219 \qquad \textbf{(E)} \ \$1449$
1985 National High School Mathematics League, 4
Given 5 points on a plane. Let $\lambda$ be the ratio of maximum value between the points to minimum value between the points. Prove that $\lambda\geq2\sin\frac{3}{10}\pi$.
2000 Korea - Final Round, 2
Prove that an $m \times n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of $8$ where $m>1$ and $n>1$
[asy]
draw ((0,0)--(0,1));
draw ((0,0)--(1.5,0));
draw ((0,1)--(.5,1));
draw ((.5,1)--(.5,0));
draw ((0,.5)--(1.5,.5));
draw ((1.5,.5)--(1.5,0));
draw ((1,.5)--(1,0));
[/asy]
2003 China Team Selection Test, 3
The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.
PEN O Problems, 10
Let $m \ge 2$ be an integer. Find the smallest integer $n>m$ such that for any partition of the set $\{m,m+1,\cdots,n\}$ into two subsets, at least one subset contains three numbers $a, b, c$ such that $c=a^{b}$.
2006 District Olympiad, 2
Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.
2009 AMC 10, 14
Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt));
path p=(1,1)--(-2,1)--(-2,2)--(1,2);
draw(p);
draw(rotate(90)*p);
draw(rotate(180)*p);
draw(rotate(270)*p);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 \plus{} \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4$
2021 Thailand TST, 2
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.
[i]Proposed by Croatia[/i]
2018 Math Prize for Girls Problems, 7
For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?
2018 Serbia National Math Olympiad, 2
Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.
2025 Ukraine National Mathematical Olympiad, 10.3
It is known that some \(d\) distinct divisors of a positive integer number \(n\) form an arithmetic progression. Prove that the number \(n\) has at least \(2d - 2\) divisors.
[i]Proposed by Anton Trygub[/i]
1997 Greece National Olympiad, 4
A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.
2000 National Olympiad First Round, 8
\[\begin{array}{rcl}
(x+y)^5 &=& z \\
(y+z)^5 &=& x \\
(z+x)^5 &=& y \end{array}\]
How many real triples $(x,y,z)$ are there satisfying above equation system?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None}
$
2015 AMC 8, 7
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{2}{9}\qquad\textbf{(C) }\frac{4}{9}\qquad\textbf{(D) }\frac{1}{2}\qquad \textbf{(E) }\frac{5}{9}$
1986 IMO Longlists, 19
Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and
\[f(x + y) - f(x) = f(x) - f(x - y)\]
for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$