Found problems: 85335
2014 AMC 12/AHSME, 2
At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $\$24.50$. How much would $8$ adult tickets and $6$ child tickets cost?
$\textbf{(A) }\$35\qquad
\textbf{(B) }\$38.50\qquad
\textbf{(C) }\$40\qquad
\textbf{(D) }\$42\qquad
\textbf{(E) }\$42.50$
2023 Vietnam Team Selection Test, 5
Let $ABCD$ be a convex quadrilateral with $\angle B < \angle A < 90^{o}$. Let $I$ be the midpoint of $AB$ and $S$ the intersection of $AD$ and $BC$. Let $R$ be a variable point inside the triangle $SAB$ such that $\angle ASR = \angle BSR$. On the straight lines $AR, BR$ , take the points $E, F$, respectively so that $BE , AF$ are parallel to $RS$. Suppose that $EF$ intersects the circumcircle of triangle $SAB$ at points $H, K$. On the segment $AB$, take points $M , N$ such that $\angle AHM =\angle BHI$ , $\angle BKN = \angle AKI$.
a) Prove that the center $J$ of the circumcircle of triangle $SMN$ lies on a fixed line.
b) On $BE, AF$ , take the points $P, Q$ respectively so that $CP$ is parallel to $SE$ and $DQ$ is parallel to $SF$. The lines $SE, SF$ intersect the circle $(SAB)$, respectively, at $U, V$. Let $G$ be the intersection of $AU$ and $BV$. Prove that the median of vertex $G$ of the triangle $GPQ$ always passes through a fixed point .
2018 Estonia Team Selection Test, 11
Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied:
1) there is at least $1$ marked point on each side,
2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$
2024 Miklos Schweitzer, 5
Let $X$ be a regular topological space and let $S$ be a countably compact dense subspace in $X$. (The countably compact property means that every infinite subset of $S$ has an accumulation point in $S$.) Show that $S$ is also $G_\delta$-dense in $X$, i.e., $S$ intersects all nonempty $G_\delta$ sets.
2024 Junior Macedonian Mathematical Olympiad, 1
Let $a, b$, and $c$ be positive real numbers. Prove that
\[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\]
When does equality hold?
[i]Proposed by Petar Filipovski[/i]
2001 South africa National Olympiad, 1
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that \[ \dfrac{1}{2}p < AC + BD < p. \] (A polygon is convex if all of its interior angles are less than $180^\circ$.)
2001 Flanders Math Olympiad, 1
may be challenge for beginner section, but anyone is able to solve it if you really try.
show that for every natural $n > 1$ we have: $(n-1)^2|\ n^{n-1}-1$
1980 IMO, 4
Let $AB$ be a diameter of a circle; let $t_1$ and $t_2$ be the tangents at $A$ and $B$, respectively; let $C$ be any point other than $A$ on $t_1$; and let $D_1D_2. E_1E_2$ be arcs on the circle determined by two lines through $C$. Prove that the lines $AD_1$ and $AD_2$ determine a segment on $t_2$ equal in length to that of the segment on $t_2$ determined by $AE_1$ and $AE_2.$
2017 Princeton University Math Competition, A5/B7
Rectangle $HOMF$ has $HO=11$ and $OM=5$. Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.
2020 Turkey Junior National Olympiad, 3
The circumcenter of an acute-triangle $ABC$ with $|AB|<|BC|$ is $O$, $D$ and $E$ are midpoints of $|AB|$ and $|AC|$, respectively. $OE$ intersects $BC$ at $K$, the circumcircle of $OKB$ intersects $OD$ second time at $L$. $F$ is the foot of altitude from $A$ to line $KL$. Show that the point $F$ lies on the line $DE$
2016 Korea Winter Program Practice Test, 2
Given an integer $n\geq 3$. For each $3\times3$ squares on the grid, call this $3\times3$ square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white.
Now suppose one can paint an infinite grid by white or black, so that one can select an $a\times b$ rectangle which contains at least $n^2-n$ isolated $3\times 3$ square. Find the minimum of $a+b$ that such thing can happen.
(Note that $a,b$ are positive reals, and selected $a\times b$ rectangle may have sides not parallel to grid line of the infinite grid.)
2021-2022 OMMC, 13
$ABCD$ is a rhombus where $\angle BAD = 60^\circ$. Point $E$ lies on minor arc $\widehat{AD}$ of the circumcircle of $ABD$, and $F$ is the intersection of $AC$ and the circle circumcircle of $EDC$. If $AF = 4$ and the circumcircle of $EDC$ has radius $14$, find the squared area of $ABCD$.
[i]Proposed by Vivian Loh [/i]
2015 Peru IMO TST, 3
Let $M$ be the midpoint of the arc $BAC$ of the circumcircle of the triangle $ABC,$ $I$ the incenter of the triangle $ABC$ and $L$ a point on the side $BC$ such that $AL$ is bisector. The line $MI$ cuts the circumcircle again at $K.$ The circumcircle of the triangle $AKL$ cuts the line $BC$ again at $P.$ Prove that $\angle AIP = 90^{\circ}.$
1966 IMO Longlists, 4
Given $5$ points in the plane, no three of them being collinear. Show that among these $5$ points, we can always find $4$ points forming a convex quadrilateral.
2019-IMOC, C4
Determine the largest $k$ such that for all competitive graph with $2019$ points, if the difference between in-degree and out-degree of any point is less than or equal to $k$, then this graph is strongly connected.
VMEO III 2006 Shortlist, N12
Given a positive integer $n > 1$. Find the smallest integer of the form $\frac{n^a-n^b}{n^c-n^d}$ for all positive integers $a,b,c,d$.
2023 Irish Math Olympiad, P2
For $n \geq 3$, a [i]special n-triangle[/i] is a triangle with $n$ distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because $41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41$, the following is a special $3$-triangle:
$$41$$
$$23\text{ }\text{ }\text{ }\text{ }\text{ }19$$
$$43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47$$
Note that a special $n$-triangle contains $3(n - 1)$ numbers.
An infinite set $A$ of positive integers is a [i]special set[/i] if, for each $n \geq 3$, the smallest $3(n - 1)$ numbers of $A$ can be used to form a special $n$-triangle.
Show that the set of positive integers that are not multiples of $2023$ is a special set.
2006 Bulgaria Team Selection Test, 1
Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and
\[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\]
for all natural $n \geq 2$.
[i]Peter Boyvalenkov[/i]
2023 OMpD, 4
Let $ABC$ be a scalene acute triangle with circumcenter $O$. Let $K$ be a point on the side $\overline{BC}$. Define $M$ as the second intersection of $\overleftrightarrow{OK}$ with the circumcircle of $BOC$. Let $L$ be the reflection of $K$ by $\overleftrightarrow{AC}$. Show that the circumcircles of the triangles $LCM$ and $ABC$ are tangent if, and only if, $\overline{AK} \perp \overline{BC}$.
2021 AIME Problems, 15
Let $f(n)$ and $g(n)$ be functions satisfying
$$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$and
$$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.3
It is known that in the triangle $ABC$, $ 2 \angle BAC + 3 \angle ABC= 180^o$. Prove that $4(BC + CA)< 5AB$.
2020 Saint Petersburg Mathematical Olympiad, 4.
Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds
$$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$
2016 Iran Team Selection Test, 3
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2010 Princeton University Math Competition, 1
PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, the probability that the 4th customer orders a previously unordered item is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2009 Today's Calculation Of Integral, 473
For nonzero real numbers $ r,\ l$ and the positive constant number $ c$, consider the curve on the $ xy$ plane : $ y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r)\quad \\
r^2 & (r\leq x\leq l \plus{} r)\quad \\
(x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r)\quad \end{array} \right.$
Denote $ V$ the volume of the solid by revolvering the curve about the $ x$ axis. Let $ r,\ l$ vary in such a way that $ r^2 \plus{} l \equal{} c$. Find the values of $ r,\ l$ which gives the maxmimum volume.