Found problems: 85335
2018 Estonia Team Selection Test, 7
Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.
1969 Canada National Olympiad, 5
Let $ABC$ be a triangle with sides of length $a$, $b$ and $c$. Let the bisector of the angle $C$ cut $AB$ in $D$. Prove that the length of $CD$ is \[ \frac{2ab\cos \frac{C}{2}}{a+b}. \]
1986 IMO Longlists, 69
Let $AX,BY,CZ$ be three cevians concurrent at an interior point $D$ of a triangle $ABC$. Prove that if two of the quadrangles $DY AZ,DZBX,DXCY$ are circumscribable, so is the third.
2008 Romania Team Selection Test, 1
Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.
Novosibirsk Oral Geo Oly IX, 2017.3
Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.
1988 Vietnam National Olympiad, 2
Suppose that $ ABC$ is an acute triangle such that $ \tan A$, $ \tan B$, $ \tan C$ are the three roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0$, where $ q\neq 1$. Show that $ p \le \minus{} 3\sqrt 3$ and $ q > 1$.
1976 Swedish Mathematical Competition, 4
A number is placed in each cell of an $n \times n$ board so that the following holds:
(A) the cells on the boundary all contain 0;
(B) other cells on the main diagonal are each1 greater than the mean of the numbers to the left and right;
(C) other cells are the mean of the numbers to the left and right.
Show that (B) and (C) remain true if ''left and right'' is replaced by ''above and below''.
2002 IMC, 11
Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$.
Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.
2001 Tournament Of Towns, 3
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.
2016 Indonesia TST, 1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
II Soros Olympiad 1995 - 96 (Russia), 11.1
Find $a$ and $b$ for which the largest and smallest is values of the function $y=\frac{x^2+ax+b}{x^2-x+1}$ are equal to the $2$ and $-3$ respectively.
2021 Sharygin Geometry Olympiad, 9.5
Let $O$ be the clrcumcenter of triangle $ABC$. Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$. Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$.
1999 IMC, 5
Suppose that $2n$ points of an $n\times n$ grid are marked. Show that for some $k > 1$ one can select $2k$ distinct marked points, say $a_1,...,a_{2k}$, such that $a_{2i-1}$ and $a_{2i}$ are in the same row, $a_{2i}$ and $a_{2i+1}$ are in the same column, $\forall i$, indices taken mod 2n.
2010 Today's Calculation Of Integral, 617
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane.
Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$
[i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]
2015 Kazakhstan National Olympiad, 4
$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.
2017 Ukraine Team Selection Test, 3
Andriyko has rectangle desk and a lot of stripes that lie parallel to sides of the desk. For every pair of stripes we can say that first of them is under second one. In desired configuration for every four stripes such that two of them are parallel to one side of the desk and two others are parallel to other side, one of them is under two other stripes that lie perpendicular to it. Prove that Andriyko can put stripes one by one such way that every next stripe lie upper than previous and get desired configuration.
[i]Proposed by Denys Smirnov[/i]
2023 Sharygin Geometry Olympiad, 9.5
A point $D$ lie on the lateral side $BC$ of an isosceles triangle $ABC$. The ray $AD$ meets the line passing through $B$ and parallel to the base $AC$ at point $E$. Prove that the tangent to the circumcircle of triangle $ABD$ at $B$ bisects $EC$.
2022 CMIMC, 2.1
A particle starts at $(0,0,0)$ in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of $\sqrt{3}$ from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of $2\sqrt{2}$ from its original location?
[i]Proposed by Connor Gordon[/i]
2009 District Olympiad, 1
On the sides $ AB $ and $ AC $ of the triangle $ ABC $ consider the points $ D, $ respectively, $ E, $ such that
$$ \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} . $$
If $ T $ is the intersection of $ DC $ and $ BE, $ determine the real number $ \alpha $ so that:
$$ \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} . $$
2018 Miklós Schweitzer, 2
A family $\mathcal{F}$ of sets is called [i]really neat[/i] if for any $A,B\in \mathcal{F}$, there is a set $C\in \mathcal{F}$ such that $A\cup B = A\cup C=B\cup C$. Let
$$f(n)=\min \left\{ \max_{A\in \mathcal{F}} |A| \colon \mathcal{F} \text{ is really neat and } |\cup \mathcal{F}| =n\right\} .$$
Prove that the sequence $f(n)/n$ converges and find its limit.
2019 CHMMC (Fall), Individual
[b]p1.[/b] Consider a cube with side length $2$. Take any one of its vertices and consider the three midpoints of the three edges emanating from that vertex. What is the distance from that vertex to the plane formed by those three midpoints?
[b]p2.[/b] Digits $H$, $M$, and $C$ satisfy the following relations where $\overline{ABC}$ denotes the number whose digits in base $10$ are $A$, $B$, and $C$.
$$\overline{H}\times \overline{H} = \overline{M}\times \overline{C} + 1$$
$$\overline{HH}\times \overline{H} = \overline{MC}\times \overline{C} + 1$$
$$\overline{HHH}\times \overline{H} = \overline{MCC}\times \overline{C} + 1$$
Find $\overline{HMC}$.
[b]p3.[/b] Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the table. If this number is even, a coin is added to the table. A player wins when he/she removes the last coin on the table. Suppose the game ends. What is the probability that the first player wins?
[b]p4.[/b] Cyclic quadrilateral $[BLUE]$ has right $\angle E$. Let $R$ be a point not in $[BLUE]$. If $[BLUR] =[BLUE]$, $\angle ELB = 45^o$, and $\overline{EU} = \overline{UR}$, find $\angle RUE$.
[b]p5.[/b] There are two tracks in the $x, y$ plane, defined by the equations
$$y =\sqrt{3 - x^2}\,\,\, \text{and} \,\,\,y =\sqrt{4- x^2}$$
A baton of length $1$ has one end attached to each track and is allowed to move freely, but no end may be picked up or go past the end of either track. What is the maximum area the baton can sweep out?
[b]p6.[/b] For integers $1 \le a \le 2$, $1 \le b \le 10$,$ 1 \le c \le 12$, $1 \le d \le 18$, let $f(a, b, c, d)$ be the unique integer between $0$ and $8150$ inclusive that leaves a remainder of a when divided by $3$, a remainder of $b$ when divided by $11$, a remainder of $c$ when divided by $13$, and a remainder of $d$ when divided by $19$. Compute $$\sum_{a+b+c+d=23}f(a, b, c, d).$$
[b]p7.[/b] Compute $\cos ( \theta)$ if $$\sum^{\infty}_{n=0} \frac{ \cos (n\theta)}{3^n} = 1.$$
[b]p8.[/b] How many solutions does this equation $$\left(\frac{a+b}{2}\right)^2=\left(\frac{b+c}{2019}\right)^2$$ have in positive integers $a, b, c$ that are all less than $2019^2$?
[b]p9.[/b] Consider a square grid with vertices labeled $1, 2, 3, 4$ clockwise in that order. Fred the frog is jumping between vertices, with the following rules: he starts at the vertex label $1$, and at any given vertex he jumps to the vertex diagonally across from him with probability $\frac12$ and the vertices adjacent to him each with probability $\frac14$ . After $2019$ jumps, suppose the probability that the sum of the labels on the last two vertices he has visited is $3$ can be written as $2^{-m} -2^{-n}$ for positive integers $m,n$. Find $m + n$.
[b]p10.[/b] The base ten numeral system uses digits $0-9$ and each place value corresponds to a power of $10$. For example, $$2019 = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0.$$
Let $\phi =\frac{1 +\sqrt5}{2}$. We can define a similar numeral system, base , where we only use digits $0$ and $1$, and each place value corresponds to a power of . For example, $$11.01 = 1 \cdot \phi^1 + 1 \cdot \phi^0 + 0 \cdot \phi^{-1} + 1 \cdot \phi^{-2}$$
Note that base representations are not unique, because, for example, $100_{\phi} = 11_{\phi}$ . Compute the base $\phi$ representation of $7$ with the fewest number of $1$s.
[b]p11.[/b] Let $ABC$ be a triangle with $\angle BAC = 60^o$ and with circumradius $1$. Let $G$ be its centroid and $D$ be the foot of the perpendicular from $A$ to $BC$. Suppose $AG =\frac{\sqrt6}{3}$ . Find $AD$.
[b]p12.[/b] Let $f(a, b)$ be a function with the following properties for all positive integers $a \ne b$:
$$f(1, 2) = f(2, 1)$$
$$f(a, b) + f(b, a) = 0$$
$$f(a + b, b) = f(b, a) + b$$
Compute: $$\sum^{2019}_{i=1} f(4^i - 1, 2^i) + f(4^i + 1, 2^i)$$
[b]p13.[/b] You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules:
1. There is a tower of height $2^n$ at the origin.
2. From towers of height $2^i \ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementioned tower and a new tower of height $2^{i-1}$. Walls must be parallel to a coordinate axis, and each tower must be connected to at least one other tower by a wall.
If one unit of tower height costs $\$9$ and one unit of wall length costs $\$3$ and $n = 1000$, how many distinct costs are there of castles that satisfy the above constraints? Two castles are distinct if there exists a tower or wall that is in one castle but not in the other.
[b]p14.[/b] For $n$ digits, $(a_1, a_2, ..., a_n)$ with $0 \le a_i < n$ for $i = 1, 2,..., n$ and $a_1 \ne 0$ define $(\overline{a_1a_2 ... a_n})_n$ to be the number with digits $a_1$, $a_2$, $...$, $a_n$ written in base $n$. Let $S_n = \{(a_1, a_2, a_3,..., a_n)| \,\,\, (n + 1)| (\overline{a_1a_2 ... a_n})_n, a_1 \ge 1\}$ be the set of $n$-tuples such that $(\overline{a_1a_2 ... a_n})_n$ is divisible by $n + 1$.
Find all $n > 1$ such that $n$ divides $|S_n| + 2019$.
[b]p15.[/b] Let $P$ be the set of polynomials with degree $2019$ with leading coefficient $1$ and non-leading coefficients from the set $C = \{-1, 0, 1\}$. For example, the function $f = x^{2019} - x^{42} + 1$ is in $P$, but the functions $f = x^{2020}$, $f = -x^{2019}$, and $f = x^{2019} + 2x^{21}$ are not in $P$.
Define a [i]swap [/i]on a polynomial $f$ to be changing a term $ax^n$ to $bx^n$ where $b \in C$ and there are no terms with degree smaller than $n$ with coefficients equal to $a$ or $b$. For example, a swap from $x^{2019} + x^{17} - x^{15} + x^{10}$ to $x^{2019} + x^{17} - x^{15} - x^{10}$ would be valid, but the following swaps would not be valid:
$$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019}$$
$$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019} + x^3 + x^2$$
$$x^{2019} + x^2 + x + 1 \,\,\, \text{to} \,\,\, x^{2019} - x^2 - x - 1$$
Let $B$ be the set of polynomials in $P$ where all non-leading terms have the same coefficient. There are $p$ polynomials that can be reached from each element of $B$ in exactly $s$ swaps, and there exist $0$ polynomials that can be reached from each element of $B$ in less than $s$ swaps.
Compute $p \cdot s$, expressing your answer as a prime factorization.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2016 Math Prize for Girls Problems, 16
Let $A < B < C < D$ be positive integers such that every three of them form the side lengths of an obtuse triangle. Compute the least possible value of $D$.
1987 Tournament Of Towns, (135) 4
We are given tiles in the form of right angled triangles having perpendicular sides of length $1$ cm and $2$ cm. Is it possible to form a square from $20$ such tiles?
( S . Fomin , Leningrad)
2024 CMIMC Integration Bee, 12
\[\int_1^\infty \frac{\sec^{-1}(x^{2})-\sec^{-1}(\sqrt x)}{x\log(x)}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2013 Uzbekistan National Olympiad, 5
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.