Found problems: 85335
1976 Euclid, 2
Source: 1976 Euclid Part B Problem 2
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Given that $x$, $y$, and $2$ are in geometric progression, and that $x^{-1}$, $y^{-1}$, and $9x^{-2}$ are in are in arithmetic progression, then find the numerical value of $xy$.
2000 Harvard-MIT Mathematics Tournament, 2
Evaluate $2000^3-1999\cdot 2000^2-1999^2\cdot 2000+1999^3$
2009 Moldova Team Selection Test, 3
[color=darkred]Quadrilateral $ ABCD$ is inscribed in the circle of diameter $ BD$. Point $ A_1$ is reflection of point $ A$ wrt $ BD$ and $ B_1$ is reflection of $ B$ wrt $ AC$. Denote $ \{P\}\equal{}CA_1 \cap BD$ and $ \{Q\}\equal{}DB_1\cap AC$. Prove that $ AC\perp PQ$.[/color]
2011 Indonesia TST, 3
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define
\[
p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.
\]
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
2024/2025 TOURNAMENT OF TOWNS, P6
An equilateral triangle is dissected into white and black triangles. It is known that all white triangles are right-angled and mutually congruent, and all black triangles are isosceles and also mutually congruent. Is it necessarily true that
a) all angles of white triangles are multiples of $30^{\circ}$; (4 marks)
b) all angles of black triangles are multiples of $30^{\circ}$ ? (5 marks)
2016 Switzerland Team Selection Test, Problem 10
Let $ABC$ be a non-rectangle triangle with $M$ the middle of $BC$. Let $D$ be a point on the line $AB$ such that $CA=CD$ and let $E$ be a point on the line $BC$ such that $EB=ED$. The parallel to $ED$ passing through $A$ intersects the line $MD$ at the point $I$ and the line $AM$ intersects the line $ED$ at the point $J$. Show that the points $C, I$ and $J$ are aligned.
2012 Gulf Math Olympiad, 2
Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\]
is $6$. For which values of $a, b$ and $c$ is equality attained?
2025 PErA, P5
We have an $n \times n$ board, filled with $n$ rectangles aligned to the grid. The $n$ rectangles cover all the board and are never superposed. Find, in terms of $n$, the smallest value the sum of the $n$ diagonals of the rectangles can take.
2015 Tournament of Towns, 6
Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs $+$ , $-$ and $\times$. He may also use a special sign $\pm$: computing the values of the resulting expression, he chooses values $+$ or $-$ for every $\pm$ in all possible combinations. For instance, the expression $5 \pm 1$ results in $\{4, 6 \}$, and $(2 \pm 0.5) \pm 0.5$ results in $\{1, 2, 3 \}$. Can Pete construct such an expression:
$a)$ If the numbers on the blackboard are $1, 2, 4$;
$b)$ For any collection of $100$ distinct real numbers on a blackboard?
2020 Princeton University Math Competition, B1
Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between $1$ and $5$ letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word! What is the maximum number of possible words in Runese?
2023 Azerbaijan IMO TST, 5
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
2012 Stanford Mathematics Tournament, 10
Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)
2016 Math Hour Olympiad, 6-7
[u]Round 1[/u]
[b]p1.[/b] At a fortune-telling exam, $13$ witches are sitting in a circle. To pass the exam, a witch must correctly predict, for everybody except herself and her two neighbors, whether they will pass or fail. Each witch predicts that each of the $10$ witches she is asked about will fail. How many witches could pass?
[b]p2.[/b] Out of $152$ coins, $7$ are counterfeit. All counterfeit coins have the same weight, and all real coins have the same weight, but counterfeit coins are lighter than real coins. How can you find $19$ real coins if you are allowed to use a balance scale three times?
[b]p3.[/b] The digits of a number $N$ increase from left to right. What could the sum of the digits of $9 \times N$ be?
[b]p4.[/b] The sides and diagonals of a pentagon are colored either blue or red. You can choose three vertices and flip the colors of all three lines that join them. Can every possible coloring be turned all blue by a sequence of such moves?
[img]https://cdn.artofproblemsolving.com/attachments/5/a/644aa7dd995681fc1c813b41269f904283997b.png[/img]
[b]p5.[/b] You have $100$ pancakes, one with a single blueberry, one with two blueberries, one with three blueberries, and so on. The pancakes are stacked in a random order. Count the number of blueberries in the top pancake and call that number $N$. Pick up the stack of the top $N$ pancakes and flip it upside down. Prove that if you repeat this counting-and-flipping process, the pancake with one blueberry will eventually end up at the top of the stack.
[u]Round 2[/u]
[b]p6.[/b] A circus owner will arrange $100$ fleas on a long string of beads, each flea on her own bead. Once arranged, the fleas start jumping using the following rules. Every second, each flea chooses the closest bead occupied by one or more of the other fleas, and then all fleas jump simultaneously to their chosen beads. If there are two places where a flea could jump, she jumps to the right. At the start, the circus owner arranged the fleas so that, after some time, they all gather on just two beads. What is the shortest amount of time it could take for this to happen?
[b]p7.[/b] The faraway land of Noetheria has $2016$ cities. There is a nonstop flight between every pair of cities. The price of a nonstop ticket is the same in both directions, but flights between different pairs of cities have different prices. Prove that you can plan a route of $2015$ consecutive flights so that each flight is cheaper than the previous one. It is permissible to visit the same city several times along the way.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1999 Harvard-MIT Mathematics Tournament, 1
Start with an angle of $60^\circ$ and bisect it, then bisect the lower $30^\circ$ angle, then the upper $15^\circ$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^\circ$ angle into two angles. Find the measure (degrees) of the smaller angle.
1984 IMO Shortlist, 13
Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume $1$ does not exceed $\frac{2}{3 \pi}.$
2024 Sharygin Geometry Olympiad, 10.3
Let $BE$ and $CF$ be the bisectors of a triangle $ABC$. Prove that $2EF \leq BF + CE$.
2020 MMATHS, I11
Let triangle $\triangle ABC$ have side lengths $AB = 7, BC = 8,$ and $CA = 9$, and let $M$ and $D$ be the midpoint of $\overline{BC}$ and the foot of the altitude from $A$ to $\overline{BC}$, respectively. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle{AEM} = m\angle{AFM} = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle{AED}$ and $\angle{AFD}$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
[i]Proposed by Andrew Wu[/i]
2015 AMC 10, 5
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place?
$\textbf{(A) } \text{David}
\qquad\textbf{(B) } \text{Hikmet}
\qquad\textbf{(C) } \text{Jack}
\qquad\textbf{(D) } \text{Rand}
\qquad\textbf{(E) } \text{Todd}
$
2016 Junior Regional Olympiad - FBH, 2
Which fraction is bigger: $\frac{5553}{5557}$ or $\frac{6664}{6669}$ ?
2003 Czech-Polish-Slovak Match, 2
In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$.
(a) Show that such a point $K$ always exists.
(b) Prove that $KD^2 = FD^2 + AF \cdot BF$.
2017 Turkey MO (2nd round), 3
Denote the sequence $a_{i,j}$ in positive reals such that $a_{i,j}$.$a_{j,i}=1$. Let $c_i=\sum_{k=1}^{n}a_{k,i}$. Prove that $1\ge$$\sum_{i=1}^{n}\dfrac {1}{c_i}$
2012 Pre - Vietnam Mathematical Olympiad, 4
Two people A and B play a game in the $m \times n$ grid ($m,n \in \mathbb{N^*}$). Each person respectively (A plays first) draw a segment between two point of the grid such that this segment doesn't contain any point (except the 2 ends) and also the segment (except the 2 ends) doesn't intersect with any other segments. The last person who can't draw is the loser. Which one (of A and B) have the winning tactics?
2010 Puerto Rico Team Selection Test, 2
There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2 \cdot a_3 \cdot ... \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$.
2023 CCA Math Bonanza, T7
The positive integer equal to the expression
\[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\]
is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.
[i]Team #7[/i]
2025 Junior Balkan Team Selection Tests - Romania, P2
Consider a scalene triangle $ABC$ with incentre $I$ and excentres $I_a,I_b,$ and $I_c$, opposite the vertices $A,B,$ and $C$ respectively. The incircle touches $BC,CA,$ and $AB$ at $E,F,$ and $G$ respectively. Prove that the circles $IEI_a,IFI_b,$ and $IGI_c$ have a common point other than $I$.