Found problems: 85335
2000 Croatia National Olympiad, Problem 2
Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Prove that the intersection of the lines $AD$ and $BE$ lies on the altitude of the triangle from $C$.
2023 Balkan MO Shortlist, C5
Find the greatest integer $k\leq 2023$ for which the following holds: whenever Alice colours exactly $k$ numbers of the set $\{1,2,\dots, 2023\}$ in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers.
Romania
2007 Estonia Math Open Senior Contests, 1
Let $ a_n \equal{} 1 \plus{} 2 \plus{} ... \plus{} n$ for every $ n \ge 1$; the numbers $ a_n$ are called triangular. Prove that if $ 2a_m \equal{} a_n$ then $ a_{2m \minus{} n}$ is a perfect square.
2016-2017 SDML (Middle School), 14
Evaluate the sum $$\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots$$
$\text{(A) }\frac{5}{9}\qquad\text{(B) }\frac{5}{8}\qquad\text{(C) }\frac{2}{3}\qquad\text{(D) }\frac{3}{4}\qquad\text{(E) }\frac{7}{9}$
2021 All-Russian Olympiad, 3
In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?
1992 AMC 12/AHSME, 14
Which of the following equations have the same graph?
$I.\ y = x - 2$
$II.\ y = \frac{x^{2} - 4}{x + 2}$
$III.\ (x + 2)y = x^{2} - 4$
$ \textbf{(A)}\ \text{I and II only} $
$ \textbf{(B)}\ \text{I and III only} $
$ \textbf{(C)}\ \text{II and III only} $
$ \textbf{(D)}\ \text{I, II and III} $
$ \textbf{(E)}\ \text{None. All equations have different graphs} $
2015 CHMMC (Fall), 9
Let $T$ be a $2015 \times 2015$ array containing the integers $1, 2, 3, ... , 2015^2$ satisfying the property that $T_{i,a }> T_{i,b}$ for all $a > b$ and $T_{c,j} > T_{d,j}$ for all $c > d$ where $1 \le a, b, c, d \le 2015$ and $T_{i,j}$ represents the entry in the $i$-th row and $j$-th column of $T$. How many possible values are there for the entry at $T_{5,5}$?
Cono Sur Shortlist - geometry, 2003.G1
Let $O$ be the circumcenter of the isosceles triangle $ABC$ ($AB = AC$). Let $P$ be a point of the segment $AO$ and $Q$ the symmetric of $P$ with respect to the midpoint of $AB$. If $OQ$ cuts $AB$ at $K$ and the circle that passes through $A, K$ and $O$ cuts $AC$ in $L$, show that $\angle ALP = \angle CLO$.
1987 AMC 12/AHSME, 13
A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to $10$ cm.)
$ \textbf{(A)}\ 36\pi \qquad\textbf{(B)}\ 45\pi \qquad\textbf{(C)}\ 60\pi \qquad\textbf{(D)}\ 72\pi \qquad\textbf{(E)}\ 90\pi $
Oliforum Contest II 2009, 3
Let a cyclic quadrilateral $ ABCD$, $ AC \cap BD \equal{} E$ and let a circle $ \Gamma$ internally tangent to the arch $ BC$ (that not contain $ D$) in $ T$ and tangent to $ BE$ and $ CE$. Call $ R$ the point where the angle bisector of $ \angle ABC$ meet the angle bisector of $ \angle BCD$ and $ S$ the incenter of $ BCE$. Prove that $ R$, $ S$ and $ T$ are collinear.
[i](Gabriel Giorgieri)[/i]
2013 Balkan MO Shortlist, A2
Let $a, b, c$ and $d$ are positive real numbers so that $abcd = \frac14$. Prove that holds
$$\left( 16ac +\frac{a}{c^2b}+\frac{16c}{a^2d}+\frac{4}{ac}\right)\left( bd +\frac{b}{256d^2c}+\frac{d}{b^2a}+\frac{1}{64bd}\right) \ge \frac{81}{4}$$
When does the equality hold?
2007 Today's Calculation Of Integral, 197
Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral.
\[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]
2016 AMC 10, 11
What is the area of the shaded region of the given $8 \times 5$ rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));
label("$1$",(1/2,5),dir(90));
label("$7$",(9/2,5),dir(90));
label("$1$",(8,1/2),dir(0));
label("$4$",(8,3),dir(0));
label("$1$",(15/2,0),dir(270));
label("$7$",(7/2,0),dir(270));
label("$1$",(0,9/2),dir(180));
label("$4$",(0,2),dir(180));
[/asy]
$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$
1984 Tournament Of Towns, (056) O4
The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?
2024 Kyiv City MO Round 2, Problem 4
Points $X$ and $Y$ are chosen inside an acute-angled triangle $ABC$ with altitude $AD$ so that $\angle BXA + \angle ACB = 180^\circ , \angle CYA + \angle ABC = 180^\circ$, and $CD + AY = BD + AX$. Point $M$ is chosen on the ray $BX$ so that $X$ lies on segment $BM$ and $XM = AC$, and point $N$ is chosen on the ray $CY$ so that $Y$ lies on segment $CN$ and $YN = AB$. Prove that $AM = AN$.
[i]Proposed by Mykhailo Shtandenko[/i]
2013 Harvard-MIT Mathematics Tournament, 2
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$?
Kvant 2023, M2760
The checkered plane is divided into dominoes. What is the maximum $k{}$ for which it is always possible to choose a $100\times 100$ checkered square containing at least $k{}$ whole dominoes?
[i]Proposed by S. Berlov[/i]
1979 IMO Longlists, 5
Describe which positive integers do not belong to the set
\[E = \left\{ \lfloor n+ \sqrt n +\frac 12 \rfloor | n \in \mathbb N\right\}.\]
1989 Tournament Of Towns, (228) 2
The hexagon $ABCDEF$ is inscribed in a circle, $AB = BC = a, CD = DE = b$, and $EF = FA = c$. Prove that the area of triangle $BDF$ equals half the area of the hexagon.
(I.P. Nagel, Yevpatoria).
2014 ASDAN Math Tournament, 2
Let $a$ and $b$ be positive integers such that $a>b$ and the difference between $a^2+b$ and $a+b^2$ is prime. Compute all possible pairs $(a,b)$.
2008 Singapore Team Selection Test, 2
Let $ x_1, x_2,\ldots , x_n$ be positive real numbers such that $ x_1x_2\cdots x_n \equal{} 1$. Prove that
\[\sum_{i \equal{} 1}^n \frac {1}{n \minus{} 1 \plus{} x_i}\le 1.\]
2015 Purple Comet Problems, 19
Let a,b,c, and d be real numbers such that \[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\] Find $1000a + 100b + 10c + d$.
1986 AIME Problems, 14
The shortest distances between an interior diagonal of a rectangular parallelepiped, P, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$.
2021 USEMO, 4
Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$.
Can line $EF$ share a point with the circumcircle of triangle $AMN?$
[i]Proposed by Sayandeep Shee[/i]
2017 Caucasus Mathematical Olympiad, 7
$10$ distinct numbers are given. Professor Odd had calculated all possible products of $1$, $3$, $5$, $7$, $9$ numbers among given numbers, and wrote down the sum of all these products. Similarly, Professor Even had calculated all possible products of $2$, $4$, $6$, $8$, $10$ numbers among given numbers, and wrote down the sum of all these products. It appears that Odd's sum is greater than Even's sum by $1$. Prove that one of $10$ given numbers is equal to $1$.