Found problems: 85335
1995 Hungary-Israel Binational, 4
Consider a convex polyhedron whose faces are triangles. Prove that it is possible to color its edges by either red or blue, in a way that the following property is satisfied: one can travel from any vertex to any other vertex while passing only along red edges, and can also do this while passing only along blue edges.
1950 AMC 12/AHSME, 45
The number of diagonals that can be drawn in a polygon of 100 sides is:
$\textbf{(A)}\ 4850 \qquad
\textbf{(B)}\ 4950\qquad
\textbf{(C)}\ 9900 \qquad
\textbf{(D)}\ 98 \qquad
\textbf{(E)}\ 8800$
2018 Costa Rica - Final Round, 2
Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, $252$, in no particular order. Determine the maximum possible value of $x + y$.
2010 Romania National Olympiad, 4
In the isosceles triangle $ABC$, with $AB=AC$, the angle bisector of $\angle B$ meets the side $AC$ at $B'$. Suppose that $BB'+B'A=BC$. Find the angles of the triangle $ABC$.
[i]Dan Nedeianu[/i]
1973 Canada National Olympiad, 6
If $A$ and $B$ are fixed points on a given circle not collinear with centre $O$ of the circle, and if $XY$ is a variable diameter, find the locus of $P$ (the intersection of the line through $A$ and $X$ and the line through $B$ and $Y$).
2004 Estonia National Olympiad, 3
From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.
2012 Purple Comet Problems, 12
Pentagon $ABCDE$ consists of a square $ACDE$ and an equilateral triangle $ABC$ that share the side $\overline{AC}$. A circle centered at $C$ has area 24. The intersection of the circle and the pentagon has half the area of the pentagon. Find the area of the pentagon.
[asy]/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(4.26cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -1.52, xmax = 2.74, ymin = -2.18, ymax = 6.72; /* image dimensions */
draw((0,1)--(2,1)--(2,3)--(0,3)--cycle);
draw((0,3)--(2,3)--(1,4.73)--cycle);
/* draw figures */
draw((0,1)--(2,1));
draw((2,1)--(2,3));
draw((2,3)--(0,3));
draw((0,3)--(0,1));
draw((0,3)--(2,3));
draw((2,3)--(1,4.73));
draw((1,4.73)--(0,3));
draw(circle((0,3), 1.44));
label("$C$",(-0.4,3.14),SE*labelscalefactor);
label("$A$",(2.1,3.1),SE*labelscalefactor);
label("$B$",(0.86,5.18),SE*labelscalefactor);
label("$D$",(-0.28,0.88),SE*labelscalefactor);
label("$E$",(2.1,0.8),SE*labelscalefactor);
/* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */[/asy]
2023 New Zealand MO, 4
Let $p$ be a prime and let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with integer coefficients such that $0 < a, b, c \le p$. Suppose $f(x)$ is divisible by $p$ whenever $x$ is a positive integer. Find all possible values of $a + b + c$.
1968 All Soviet Union Mathematical Olympiad, 112
The circle inscribed in the triangle $ABC$ touches the side $[AC]$ in the point $K$. Prove that the line connecting the midpoint of the side $[AC]$ with the centre of the circle halves the segment $[BK]$ .
1987 AMC 8, 13
Which of the following fractions has the largest value?
$\text{(A)}\ \frac{3}{7} \qquad \text{(B)}\ \frac{4}{9} \qquad \text{(C)}\ \frac{17}{35} \qquad \text{(D)}\ \frac{100}{201} \qquad \text{(E)}\ \frac{151}{301}$
1980 Vietnam National Olympiad, 2
Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that
\[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]
2016 Azerbaijan Balkan MO TST, 3
$k$ is a positive integer. $A$ company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself $;$ it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least $k$ customers, this person gets a present. Prove that, if $n$ persons have bought clocks, then at most $\frac{n}{k+2}$ presents have been accepted.
2011 Pre-Preparation Course Examination, 2
prove that $\pi_1 (X,x_0)$ is not abelian. $X$ is like an eight $(8)$ figure.
[b]comments:[/b] eight figure is the union of two circles that have one point $x_0$ in common.
we call a group $G$ abelian if: $\forall a,b \in G:ab=ba$.
2011 IFYM, Sozopol, 3
Let $a=x_1\leq x_2\leq ...\leq x_n=b$. Prove the following inequality:
$(x_1+x_2+...+x_n )(\frac{1}{x_1} +\frac{1}{x_2} +...+\frac{1}{x_n} )\leq \frac{(a+b)}{4ab} n^2$.
2000 AMC 12/AHSME, 20
If $ x$, $ y$, and $ z$ are positive numbers satisfying \[x \plus{} 1/y \equal{} 4,\quad y \plus{} 1/z \equal{} 1,\quad\text{and}\quad z \plus{} 1/x \equal{} 7/3,\] then $ xyz \equal{}$
$ \textbf{(A)}\ 2/3 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 7/3$
2025 Harvard-MIT Mathematics Tournament, 12
Holden has a collection of polygons. He writes down a list containing the measure of each interior angle of each of his polygons. He writes down the list $30^\circ, 50^\circ, 60^\circ, 70^\circ, 90^\circ, 100^\circ, 120^\circ, 160^\circ,$ and $x^\circ,$ in some order. Compute $x.$
Today's calculation of integrals, 852
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows.
(1) $g_n(x)=(1+x)^n$
(2) $g_n(x)=\sin n\pi x$
(3) $g_n(x)=e^{nx}$
2007 Bulgarian Autumn Math Competition, Problem 12.4
Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by:
\[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\]
Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.
1999 Switzerland Team Selection Test, 10
Prove that the product of five consecutive positive integers cannot be a perfect square.
2001 Manhattan Mathematical Olympiad, 4
You have a pencil, paper and an angle of $19$ degrees made out of two equal very thin sticks. Can you construct an angle of $1$ degree using only these tools?
2002 China Team Selection Test, 2
For any two rational numbers $ p$ and $ q$ in the interval $ (0,1)$ and function $ f$, there is always $ \displaystyle f \left( \frac{p\plus{}q}{2} \right) \leq \frac{f(p) \plus{} f(q)}{2}$. Then prove that for any rational numbers $ \lambda, x_1, x_2 \in (0,1)$, there is always:
\[ f( \lambda x_1 \plus{} (1\minus{}\lambda) x_2 ) \leq \lambda f(x_i) \plus{} (1\minus{}\lambda) f(x_2)\]
2018 CMIMC CS, 8
We consider a simple model for balanced parenthesis checking. Let $\mathcal R=\{\texttt{(())}\rightarrow \texttt{A},\texttt{(A)}\rightarrow\texttt{A},\texttt{AA}\rightarrow\texttt{A}\}$ be a set of rules for phrase reduction. Ideally, any given phrase is balanced if and only if the model is able to reduce the phrase to $\texttt{A}$ by some arbitrary sequence of rule applications. For example, to show $\texttt{((()))}$ is balanced we can perform the following sequence of reductions.
\[\texttt{((()))}\rightarrow\texttt{(A)}\rightarrow\texttt{A}\qquad \checkmark\]
Unfortunately, the above set of rules $\mathcal R$ is not complete, since there exist parenthetical phrases which are balanced but which are not balanced according to $\mathcal R$. Determine the number of such phrases of length $14$.
2024 Putnam, B5
Let $k$ and $m$ be positive integers. For a positive integer $n$, let $f(n)$ be the number of integer sequences $x_1,\,\ldots,\,x_k,\,y_1,\,\ldots,\,y_m,\,z$ satisfying $1\leq x_1\leq\cdots\leq x_k\leq z\leq n$ and $1\leq y_1\leq\cdots\leq y_m\leq z\leq n$. Show that $f(n)$ can be expressed as a polynomial in $n$ with nonnegative coefficients.
Kvant 2022, M2711
Three pairwise externally tangent circles $\omega_1,\omega_2$ and $\omega_3$ are given. Let $K_{12}$ be the point of tangency between $\omega_1$ and $\omega_2$ and define $K_{23}$ and $K_{31}$ similarly. Consider the point $A_1$ on $\omega_1$. Let $A_2$ be the second intersection of the line $A_1K_{12}$ with $\omega_2$. The line $A_2K_{23}$ then intersects $\omega_3$ the second time at $A_3$, and then line $A_3K_{31}$ intersects $\omega_1$ again at $A_4$ and so on.
[list=a]
[*]Prove that after six steps, the process will loop; that is, $A_7=A_1$.
[*]Prove that the lines $A_1A_2$ and $A_4A_5$ are perpendicular.
[*]Prove that the triples of lines $A_1A_2,A_3A_4$ and $A_5A_6$ and $A_2A_3,A_4A_5$ and $A_6A_1$ intersect at two diametrically opposite points on the circle $(K_{12}K_{23}K_{31})$.
[/list]
[i]Proposed by E. Morozov[/i]
2019 Jozsef Wildt International Math Competition, W. 1
The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\geq 2$. Find $$\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}$$