Found problems: 85335
1992 Flanders Math Olympiad, 4
Let $A,B,P$ positive reals with $P\le A+B$.
(a) Choose reals $\theta_1,\theta_2$ with $A\cos\theta_1 + B\cos\theta_2=P$ and prove that \[ A\sin\theta_1 + B\sin\theta_2 \le \sqrt{(A+B-P)(A+B+P)} \]
(b) Prove equality is attained when $\theta_1=\theta_2=\arccos\left(\dfrac{P}{A+B}\right)$.
(c) Take $A=\dfrac{1}{2}xy, B=\dfrac{1}{2}wz$ and $P=\dfrac14 \left(x^2+y^2-z^2-w^2\right)$ with $0<x\le y\le x+z+w$, $z,w>0$ and $z^2+w^2<x^2+y^2$.
Show that we can translate (a) and (b) into the following theorem: from all quadrilaterals with (ordered) sidelenghts $(x,y,z,w)$, the cyclical one has the greatest area.
IV Soros Olympiad 1997 - 98 (Russia), 11.6
There are $6$ points marked on the plane. Find the greatest possible number of acute triangles with vertices at the marked points.
2014 Contests, 3
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
2006 National Olympiad First Round, 32
What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set $\{1,2,\dots, 9\}$, there exist $k$ of them having at least one common element." true?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2020 AIME Problems, 4
Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, and $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$, where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.
1999 Brazil Team Selection Test, Problem 2
If $a,b,c,d$ are Distinct Real no. such that
$a = \sqrt{4+\sqrt{5+a}}$
$b = \sqrt{4-\sqrt{5+b}}$
$c = \sqrt{4+\sqrt{5-c}}$
$d = \sqrt{4-\sqrt{5-d}}$
Then $abcd = $
1991 Arnold's Trivium, 60
Is there a solution of the Cauchy problem
\[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\]
in a neighbourhood of the point $(x_0,0)$ of the $x$-axis? Is it unique?
Estonia Open Senior - geometry, 2010.1.4
Circle $c$ passes through vertices $A$ and $B$ of an isosceles triangle $ABC$, whereby line $AC$ is tangent to it. Prove that circle $c$ passes through the circumcenter or the incenter or the orthocenter of triangle $ABC$.
2005 Tuymaada Olympiad, 7
Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$.
[i]Proposed by S. Berlov[/i]
2002 Croatia National Olympiad, Problem 3
If two triangles with side lengths $a,b,c$ and $a',b',c'$ and the corresponding angle $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$ satisfy $\alpha+\alpha'=\pi$ and $\beta=\beta'$, prove that $aa'=bb'+cc'$.
2023 BMT, 16
Sabine rolls a fair $14$-sided die numbered $1$ to $14$ and gets a value of $x$. She then draws $x$ cards uniformly at random (without replacement) from a deck of $14$ cards, each of which labeled a different integer from $1$ to $14$. She finally sums up the value of her die roll and the value on each card she drew to get a score of $S$. Let $A$ be the set of all obtainable scores. Compute the probability that $S$ is greater than or equal to the median of $A$.
1998 Korea - Final Round, 3
Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.
1973 AMC 12/AHSME, 32
The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is
$ \textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 9/2 \qquad
\textbf{(C)}\ 27/2 \qquad
\textbf{(D)}\ \frac{9\sqrt3}{2} \qquad
\textbf{(E)}\ \text{none of these}$
2005 Cono Sur Olympiad, 2
We say that a number of 20 digits is [i]special[/i] if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.
2017 AMC 8, 22
In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
[asy] draw((0,0)--(12,0)--(12,5)--(0,0)); draw(arc((8.67,0),(12,0),(5.33,0))); label("$A$", (0,0), W); label("$C$", (12,0), E); label("$B$", (12,5), NE); label("$12$", (6, 0), S); label("$5$", (12, 2.5), E);[/asy]
$\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{13}{5}\qquad\textbf{(C) }\frac{59}{18}\qquad\textbf{(D) }\frac{10}{3}\qquad\textbf{(E) }\frac{60}{13}$
2025 Sharygin Geometry Olympiad, 22
A circle and an ellipse with foci $F_{1}$, $F_{2}$ lying inside it are given. Construct a chord $AB$ of the circle touching the ellipse and such that $AF_{1}F_{2}B$ is a cyclic quadrilateral.
Proposed by: A.Zaslavsky
Mathley 2014-15, 5
A quadrilateral $ABCD$ is inscribed in a circle $(O)$. Another circle $(I)$ is tangent to the diagonals $AC, BD$ at $M, N$ respectively. Suppose that $MN$ meets $AB,CD$ at $P, Q$ respectively. The circumcircle of triangle $IMN$ meets the circumcircles of $IAB, ICD$ at $K, L$ respectively, which are distinct from $I$. Prove that the lines $PK, QL$, and $OI$ are concurrent.
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
2014 Contests, 1
Let $p$ be a prime such that $p\mid 2a^2-1$ for some integer $a$. Show that there exist integers $b,c$ such that $p=2b^2-c^2$.
2023 Caucasus Mathematical Olympiad, 8
Let $ABC$ be an equilateral triangle with the side length equals $a+ b+ c$. On the side $AB{}$ of the triangle $ABC$ points $C_1$ and $C_2$ are chosen, on the side $BC$ points $A_1$ and $A_2$, arc chosen, and on the side $CA$ points $B_1$ and $B_2$ are chosen such that $A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b,
C_1C_2 = BA_1 = AB_2 = c$. Let the point $A^{’}$ be such that the triangle $A^{'} B_2C_1$ is equilateral, and the points $A$ and $A^{'}$ lie on different sides of the line $B_2C_1$. Similarly, the points $B^{’}$ and $C^{'}$ are constructed (the triangle $B^{'} C_2A_1$ is equilateral, and the points $B$ and $B^{’}$ lie on different sides of the line $C_2A_1$; the triangle $C^{'} A_2B_1$ is equilateral, and the points $C$ and $C^{'}$ lie on different sides of the line $A_2B_1$). Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.
1998 IMC, 1
Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \subseteq U_2, \dim U_1 =3, \dim U_2=6$. Let $\varepsilon$ be the set of all linear maps $T: V\rightarrow V$ which have $T(U_1)\subseteq U_1, T(U_2)\subseteq U_2$. Calculate the dimension of $\varepsilon$. (again, all as real vector spaces)
Kyiv City MO 1984-93 - geometry, 1988.9.1
Each side of a convex quadrilateral is less than $20$ cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point $Q$ inside the quadrilateral is less than $15$ cm.
2009 Turkey Team Selection Test, 3
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
2012 Denmark MO - Mohr Contest, 4
Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.
2020 Peru Iberoamerican Team Selection Test, P7
The numbers $1, 2,\ldots ,50$ are written on a blackboard. Ana performs the following operations: she chooses any three numbers $a, b$ and $c$ from the board and replaces them with their sum $a + b + c$ and writes the number $(a + b) (b + c) (c + a)$ in the notebook. Ana performs these operations until there are only two numbers left on the board ($24$ operations in total). Then, she calculates the sum of the numbers written down in her notebook. Let $M$ and $m$ be the maximum and minimum possible of the sums obtained by Ana.
Find the value of $\frac{M}{m}$.
2017 China Western Mathematical Olympiad, 4
Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?