Found problems: 85335
2015 Moldova Team Selection Test, 3
Consider an acute triangle $ABC$, points $E,F$ are the feet of the perpendiculars from $B$ and $C$ in $\triangle ABC$. Points $I$ and $J$ are the projections of points $F,E$ on the line $BC$, points $K,L$ are on sides $AB,AC$ respectively such that $IK \parallel AC$ and $JL \parallel AB$. Prove that the lines $IE$,$JF$,$KL$ are concurrent.
1976 Czech and Slovak Olympiad III A, 6
Consider two non-parallel half-planes $\pi,\pi'$ with the common boundary line $p.$ Four different points $A,B,C,D$ are given in the half-plane $\pi.$ Similarly, four points $A',B',C',D'\in\pi'$ are given such that $AA'\parallel BB'\parallel CC'\parallel DD'$. Moreover, none of these points lie on $p$ and the points $A,B,C,D'$ form a tetrahedron. Show that the points $A',B',C',D$ also form a tetrahedron with the same volume as $ABCD'.$
2021 Malaysia IMONST 1, 10
Determine the number of integer solutions $(x, y, z)$, with $0 \le x, y, z \le 100$, for the equation $$(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.$$
1999 USAMTS Problems, 4
We say a triangle in the coordinate plane is [i]integral[/i] if its three vertices have integer coordinates and if its three sides have integer lengths.
(a) Find an integral triangle with perimeter of $42$.
(b) Is there an integral triangle with perimeter of $43$?
2017 NIMO Problems, 3
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
[i]Proposed by Anand Iyer[/i]
2000 Abels Math Contest (Norwegian MO), 3
a) Each point, on the perimeter of a square, is colored either red, or blue. Show that, there is a right-angled triangle where all the corners are on the square of the square and so that all the corners are on points of the same color.
b) Show that, it is possible to color each point on the perimeter of one square, red, white, or blue so that, there is not a right-angled triangle where all the three corners are at points of same color.
2007 Grigore Moisil Intercounty, 3
Let be two functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ g $ has infinite limit at $ \infty . $
[b]a)[/b] Prove that if $ g $ continuous then $ \lim_{x\to\infty } f(x) =\lim_{x\to\infty } f(g(x)) $
[b]b)[/b] Provide an example of what $ f,g $ could be if $ f $ has no limit at $ \infty $ and $ \lim_{x\to\infty } f(g(x)) =0. $
2011 Lusophon Mathematical Olympiad, 3
Consider a sequence of equilateral triangles $T_{n}$ as represented below:
[asy]
defaultpen(linewidth(0.8));size(350);
real r=sqrt(3);
path p=origin--(2,0)--(1,sqrt(3))--cycle;
int i,j,k;
for(i=1; i<5; i=i+1) {
for(j=0; j<i; j=j+1) {
for(k=0; k<j; k=k+1) {
draw(shift(5*i-5+(i-2)*(i-1)*1,0)*shift(2(j-k)+k, k*r)*p);
}}}[/asy]
The length of the side of the smallest triangles is $1$. A triangle is called a delta if its vertex is at the top; for example, there are $10$ deltas in $T_{3}$. A delta is said to be perfect if the length of its side is even. How many perfect deltas are there in $T_{20}$?
2023 Math Prize for Girls Olympiad, 1
Let $n \ge 2023$ be an integer. Prove that there exists a permutation $(p_1, p_2, \dots, p_n)$ of $(1, 2, \dots, n)$ such that
\[
p_1 + 2p_2 + 3p_3 + \dots + np_n
\]
is divisible by $n$.
2013 Nordic, 1
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
2013 Tournament of Towns, 2
A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.
2023 Bulgarian Spring Mathematical Competition, 9.2
Given is triangle $ABC$ with angle bisector $CL$ and the $C-$median meets the circumcircle $\Gamma$ at $D$. If $K$ is the midpoint of arc $ACB$ and $P$ is the symmetric point of $L$ with respect to the tangent at $K$ to $\Gamma$, then prove that $DLCP$ is cyclic.
1984 All Soviet Union Mathematical Olympiad, 388
The $A,B,C$ and $D$ points (from left to right) belong to the straight line. Prove that every point $E$, that doesn't belong to the line satisfy: $$|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|$$
1996 Singapore Team Selection Test, 1
Let $P$ be a point on the side $AB$ of a square $ABCD$ and $Q$ a point on the side $BC$. Let $H$ be the foot of the perpendicular from $B$ to $PC$. Suppose that $BP = BQ$. Prove that $QH$ is perpendicular to $HD$.
2020 CHKMO, 2
Let $S={1,2,\ldots,100}$. Consider a partition of $S$ into $S_1,S_2,\ldots,S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $\displaystyle S=\bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]
Among all $n$ and partitions of $S$, determine the minimum possible score.
2014 India PRMO, 20
What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$?
1994 Brazil National Olympiad, 3
We are given n objects of identical appearance, but different mass, and a balance which can be used to compare any two objects (but only one object can be placed in each pan at a time). How many times must we use the balance to find the heaviest object and the lightest object?
1991 Baltic Way, 4
A polynomial $p$ with integer coefficients is such that $p(-n) < p(n) < n$ for some integer $n$. Prove that $p(-n) < -n$.
2023 Mongolian Mathematical Olympiad, 3
Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is [i] good [/i], if the sum of any $m$ consecutive numbers on this circle is a power of $m$.
1. Let $n \geq 2$ be a positive integer. Prove that for any [i] good [/i] sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a [i] good [/i] sequence.
2. Prove that in any [i] good [/i] sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.
1964 AMC 12/AHSME, 21
If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals:
$ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $
2013 Princeton University Math Competition, 15
Prove: \[|\sin a_1|+|\sin a_2|+|\sin a_3|+\ldots+|\sin a_n|+|\cos(a_1+a_2+a_3+\ldots+a_n)|\geq 1.\]
2008 Singapore Junior Math Olympiad, 2
Let $a.b,c,d$ be positive real numbers such that $cd = 1$. Prove that there is an integer $n$ such that $ab\le n^2\le (a + c)(b + d)$.
2007 Stanford Mathematics Tournament, 4
Evaluate $ (\tan 10^\circ)(\tan 20^\circ)(\tan 30^\circ)(\tan 40^\circ)(\tan 50^\circ)(\tan 60^\circ)(\tan 70^\circ)(\tan 80^\circ)$.
2023 Baltic Way, 15
Let $\omega_1$ and $\omega_2$ be two circles with no common points, such that any of them is not inside the other one. Let $M, N$ lie on $\omega_1, \omega_2$, such that the tangents at $M$ to $\omega_1$ and $N$ to $\omega_2$ meet at $P$, such that $PM=PN$. The circles $\omega_1$, $\omega_2$ meet $MN$ at $A, B$. The lines $PA, PB$ meet $\omega_1, \omega_2$ at $C, D$. Show that $\angle BCN=\angle ADM$.
2020 Yasinsky Geometry Olympiad, 4
Let $BB_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. From the point $B_1$ the perpendiculars $B_1E$ and $B_1F$ are drawn on the sides $AB$ and $BC$ of the triangle, respectively, and from the point $C_1$ the perpendiculars $C_1 K$ and $C_1L$ on the sides $AC$ and $BC$, respectively. It turned out that the lines $EF$ and $KL$ are perpendicular. Find the measure of the angle $A$ of the triangle $ABC$.
(Alexander Dunyak)