Found problems: 85335
1982 IMO Longlists, 47
Evaluate $\sec'' \frac{\pi}4 +\sec'' \frac{3\pi}4+\sec'' \frac{5\pi}4+\sec'' \frac{7\pi}4$. (Here $\sec''$ means the second derivative of $\sec$).
2004 Unirea, 4
The circles $ C_1,C_2 $ meet at the points $ A,B. $ A line thru $ A $ intersects $ C_1,C_2 $ at $ C,D, $ respectively. Point $ A $ is not on the arc $ BC $ of $ C_1, $ neither on the arc $ BD $ of $ C_2. $ On the segments $ CD,BC,BD $ there are the points $ M,N,K $ such that $ MN $ is parallel to $ BD $ and $ MK $ is parallel with $ BC. $ Upon the arc $ BC $ let $ E $ be a point having the property that $ EN $ is perpendicular to $ BC, $ and upon the arc $ BD $ let $ F $ be a point chosen so that $ FK $ is perpendicular to $ BD. $ Show that the angle $ \angle EMF $ is right.
2013 Moldova Team Selection Test, 2
Let $a_n=1+n!(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!})$ for any $n\in \mathbb{Z}^{+}$. Consider $a_n$ points in the plane,no $3$ of them collinear.The segments between any $2$ of them are colored in one of $n$ colors. Prove that among them there exist $3$ points forming a monochromatic triangle.
2021 Romania National Olympiad, 4
Let $n \ge 2$ and matrices $A,B \in M_n(\mathbb{R})$. There exist $x \in \mathbb{R} \backslash \{0,\frac{1}{2}, 1 \}$, such that $ xAB + (1-x)BA = I_n$. Show that $(AB-BA)^n = O_n$.
2019 India Regional Mathematical Olympiad, 4
Let $a_1,a_2,\cdots,a_6,a_7$ be seven positive integers. Let $S$ be the set of all numbers of the form $a_i^2+a_j^2$ where $1\leq i<j\leq 7$.
Prove that there exist two elements of $S$ which have the same remainder on dividing by $36$.
1995 AMC 8, 25
Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes $5$ hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass in the highway (not in the station)?
$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 11$
2014 Germany Team Selection Test, 2
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$.
Prove
\[ AN \cdot NC = CD \cdot BN. \]
Ukraine Correspondence MO - geometry, 2006.10
Let $ABC$ be an isosceles triangle ($AB=AC$). An arbitrary point $M$ is chosen on the extension of the $BC$ beyond point $B$. Prove that the sum of the radius of the circle inscribed in the triangle $AMB$ and the radius of the circle tangent to the side $AC$ and the extensions of the sides $AM, CM$ of the triangle $AMC$ does not depend on the choice of point $M$.
2020 CMIMC Team, 7
Points $P$ and $Q$ lie on a circle $\omega$. The tangents to $\omega$ at $P$ and $Q$ intersect at point $T$, and point $R$ is chosen on $\omega$ so that $T$ and $R$ lie on opposite sides of $PQ$ and $\angle PQR = \angle PTQ$. Let $RT$ meet $\omega$ for the second time at point $S$. Given that $PQ = 12$ and $TR = 28$, determine $PS$.
2014 Taiwan TST Round 2, 1
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$
Prove that $u_n = v_n.$
2004 Brazil National Olympiad, 3
Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?
2002 AMC 10, 8
How many ordered triples of positive integers $(x,y,z)$ satisfy $(x^y)^z=64$?
$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
1997 All-Russian Olympiad Regional Round, 8.8
In Mexico City, to limit traffic flow, each private car is set two days of the week on which it cannot go out onto the city streets. A family needs to have at least 10 cars at its disposal every day. What is the smallest number of cars can get by with the seven if its members can choose forbidden days for your cars?
2017 Azerbaijan Junior National Olympiad, P3
Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.
2014 BMT Spring, 4
Alice, Bob, Cindy, David, and Emily sit in a circle. Alice refuses to sit to the right of Bob, and Emily sits next to Cindy. If David sits next to two girls, determine who could sit immediately to the right of Alice.
2020 Novosibirsk Oral Olympiad in Geometry, 2
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
1977 IMO Shortlist, 9
For which positive integers $n$ do there exist two polynomials $f$ and $g$ with integer coefficients of $n$ variables $x_1, x_2, \ldots , x_n$ such that the following equality is satisfied:
\[\sum_{i=1}^n x_i f(x_1, x_2, \ldots , x_n) = g(x_1^2, x_2^2, \ldots , x_n^2) \ ? \]
LMT Team Rounds 2010-20, A1 B9
Ben writes the string $$\underbrace{111\ldots 11}_{2020 \text{ digits}}$$on a blank piece of paper. Next, in between every two consecutive digits, he inserts either a plus sign $(+)$ or a multiplication sign $(\times)$. He then computes the expression using standard order of operations. Find the number of possible distinct values that Ben could have as a result.
[i]Proposed by Taiki Aiba[/i]
1997 Italy TST, 2
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
1973 AMC 12/AHSME, 5
Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),
I. Averaging is associative
II. Averaging is commutative
III. Averaging distributes over addition
IV. Addition distributes over averaging
V. Averaging has an identity element
those which are always true are
$ \textbf{(A)}\ \text{All} \qquad
\textbf{(B)}\ \text{I and II only} \qquad
\textbf{(C)}\ \text{II and III only} \qquad
\textbf{(D)}\ \text{II and IV only} \qquad
\textbf{(E)}\ \text{II and V only}$
2023 German National Olympiad, 2
In a triangle, the edges are extended past both vertices by the length of the edge opposite to the respective vertex.
Show that the area of the resulting hexagon is at least $13$ times the area of the original triangle.
2009 Mexico National Olympiad, 3
Let $a$, $b$, and $c$ be positive numbers satisfying $abc=1$. Show that
\[\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1\]
1961 IMO Shortlist, 5
Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b \] In what case does the equality hold?
III Soros Olympiad 1996 - 97 (Russia), 9.4
At what values of $a$ does each of the equations $x^2 + ax + 1996 = 0$ or $x^2 + 1996x + a = 0$ have two integer roots?
2006 Moldova National Olympiad, 11.4
On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.