Found problems: 85335
1962 AMC 12/AHSME, 23
In triangle $ ABC$, $ CD$ is the altitude to $ AB$ and $ AE$ is the altitude to $ BC.$ If the lengths of $ AB, CD,$ and $ AE$ are known, the length of $ DB$ is:
$ \textbf{(A)}\ \text{not determined by the information given} \qquad$
$ \textbf{(B)}\ \text{determined only if A is an acute angle} \qquad$
$ \textbf{(C)}\ \text{determined only if B is an acute angle} \qquad$
$ \textbf{(D)}\ \text{determined only in ABC is an acute triangle} \qquad$
$ \textbf{(E)}\ \text{none of these is correct}$
Kyiv City MO Juniors 2003+ geometry, 2010.89.4
Point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AO$ intersects the side $BC$ at point $D$ so that $OD = BD = 1/3 BC$ . Find the angles of the triangle $ABC$. Justify the answer.
2021 Latvia Baltic Way TST, P5
Six lines are drawn in the plane. Determine the maximum number of points, through which at least $3$ lines pass.
1998 VJIMC, Problem 4-M
A function $f:\mathbb R\to\mathbb R$ has the property that for every
$x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such
that $0<t<1$ and
$$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$
Does it imply that
$$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$
for every $x,y\in\mathbb R$?
2021 Brazil Undergrad MO, Problem 3
Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.
2024 AMC 10, 4
Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposted in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22,23,\ldots,28$ are despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?
$\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$
2020 CMIMC Team, 14
Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?
2014 Lithuania Team Selection Test, 3
Given such positive real numbers $a, b$ and $c$, that the system of equations:
$ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $
has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.
2021 CCA Math Bonanza, L1.3
A coin is flipped $20$ times. Let $p$ be the probability that each of the following sequences of flips occur exactly twice:
[list]
[*] one head, two tails, one head
[*] one head, one tails, two heads.
[/list]
Given that $p$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $\gcd (m,n)$.
[i]2021 CCA Math Bonanza Lightning Round #1.3[/i]
2013 Junior Balkan MO, 4
Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses $n$ real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial $n$ numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?
a. $n=5$
b. $n=6$
c. $n=8$
Justify your answer(s).
[For example, when $n=4$, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]
2019 CMIMC, 4
Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ such that $C_1$ is the midpoint of $\overline{A_2B_1}$. She performs analogous operations on $B_2$ and $C_2$. What is the expected value of the area of $\triangle A_2B_2C_2$?
2013 Philippine MO, 2
2. Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .
2009 ISI B.Stat Entrance Exam, 3
Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR, APQ$ and $PQCR$. Find the minimum possible value of $M$.
2011 Bosnia And Herzegovina - Regional Olympiad, 4
Let $n$ be a positive integer and set $S=\{n,n+1,n+2,...,5n\}$
$a)$ If set $S$ is divided into two disjoint sets , prove that there exist three numbers $x$, $y$ and $z$(possibly equal) which belong to same subset of $S$ and $x+y=z$
$b)$ Does $a)$ hold for set $S=\{n,n+1,n+2,...,5n-1\}$
2018 Middle European Mathematical Olympiad, 3
Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A.$ Let $R$ and $Q$ be the centroids of triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and points $P$ $Q$ $R$ and $D$ are concyclic .Prove that the lines $AP$ $BQ$ and $CR$ are concurrent.
KoMaL A Problems 2020/2021, A. 783
A polyomino is a figure which consists of unit squares joined together by their sides. (A polyomino may contain holes.) Let $n\ge3$ be a positive integer. Consider a grid of unit square cells which extends to infinity in all directions. Find, in terms of $n$, the greatest positive integer $C$ which satisfies the following condition: For every colouring of the cells of the grid in $n$ colours, there is some polyomino within the grid which contains at most $n-1$ colours and whose area is at least $C$.
Proposed by Nikolai Beluhov, Stara Zagora, Bulgaria and Stefan Gerdjikov, Sofia, Bulgaria
2018 India IMO Training Camp, 2
Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.
2005 AIME Problems, 14
Consider the points $A(0,12)$, $B(10,9)$, $C(8,0)$, and $D(-4,7)$. There is a unique square $S$ such that each of the four points is on a different side of $S$. Let $K$ be the area of $S$. Find the remainder when $10K$ is divided by $1000$.
2017 German National Olympiad, 1
Given two real numbers $p$ and $q$, we study the following system of equations with variables $x,y \in \mathbb{R}$:
\begin{align*} x^2+py+q&=0,\\
y^2+px+q&=0.
\end{align*}
Determine the number of distinct solutions $(x,y)$ in terms of $p$ and $q$.
2017 Canada National Olympiad, 1
For pairwise distinct nonnegative reals $a,b,c$, prove that
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.
1981 Czech and Slovak Olympiad III A, 5
Let $n$ be a positive integer. Determine the maximum of the sum $x_1+\cdots+x_n$ where $x_1,\ldots,x_n$ are non-negative integers satisfying the condition \[x_1^3+\cdots+x_n^3\le7n.\]
2022 BMT, 17
Compute the number of ordered triples $(a, b, c)$ of integers between $-100$ and $100$ inclusive satisfying the simultaneous equations
$$a^3 - 2a = abc - b - c$$
$$b^3 - 2b = bca - c - a$$
$$c^3 - 2c = cab - a - b.$$
2010 National Chemistry Olympiad, 22
Which reaction has the most positive entropy change under standard conditions?
$ \textbf{(A)}\hspace{.05in}\ce{H2O}_{(g)}+\ce{CO}_{(g)} \rightarrow \ce{H2}_{(g)}+ \ce{CO2}_{(g)}\qquad$
$\textbf{(B)}\hspace{.05in}\ce{CaCO3}_{(s)} \rightarrow \ce{CaO}_{(s)} + \ce{CO2}_{(g)} \qquad$
$\textbf{(C)}\hspace{.05in}\ce{NH3}_{(g)} \rightarrow \ce{NH3}_{(aq)}\qquad$
$\textbf{(D)}\hspace{.05in}\ce{C8H18}_{(l)} \rightarrow \ce{C8H18}_{(s)}\qquad$
2023 HMNT, 11
Let $ABCD$ and $WXYZ$ be two squares that share the same center such that $WX \parallel AB$ and $WX < AB.$ Lines $CX$ and $AB$ intersect at $P,$ and lines $CZ$ and $AD$ intersect at $Q.$ If points $P, W,$ and $Q$ are collinear, compute the ratio $AB/WX.$
1989 Vietnam National Olympiad, 1
Are there integers $ x$, $ y$, not both divisible by $ 5$, such that $ x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}$?