Found problems: 85335
Champions Tournament Seniors - geometry, 2008.4
Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.
2001 Swedish Mathematical Competition, 1
Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same:
4 6 10 14 22 26
6 9 15 21 33 39
10 15 25 35 55 65
16 24 40 56 88 104
18 27 45 63 99 117
20 30 50 70 110 130
2022 Yasinsky Geometry Olympiad, 6
Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$.
(Matvii Kurskyi)
2017 F = ma, 11
A small hard solid sphere of mass m and negligible radius is connected to a thin rod of length L and
mass 2m. A second small hard solid sphere, of mass M and negligible radius, is fired perpendicularly
at the rod at a distance h above the sphere attached to the rod, and sticks to it.
A: h = 0
B: h = L/3
C: h = L/2
D: h = L
E: Any L will work
2007 AIME Problems, 12
In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at $(20, 0)$. Point $C$ is in the first quadrant with $AC = BC$ and $\angle BAC = 75^\circ$. If $\triangle ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s$ where $p$, $q$, $r$, $s$ are integers. Find $(p-q+r-s)/2$.
2015 IFYM, Sozopol, 2
Let $ABCD$ be an inscribed quadrilateral and $P$ be an inner point for it so that $\angle PAB=\angle PBC=\angle PCD=\angle PDA$. The lines $AD$ and $BC$ intersect in point $Q$ and lines $AB$ and $CD$ – in point $R$. Prove that $\angle (PQ,PR)=\angle (AC,BD)$.
1991 China Team Selection Test, 1
We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$
1991 French Mathematical Olympiad, Problem 1
(a) Suppose that $x_n~(n\ge0)$ is a sequence of real numbers with the property that $x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2$ for each $n\in\mathbb N$. Prove that for each $n\in\mathbb N_0$ there exists $m\in\mathbb N_0$ such that $x_0+x_1+\ldots+x_n=\frac{m(m+1)}2$.
(b) For natural numbers $n$ and $p$, we define $S_{n,p}=1^p+2^p+\ldots+n^p$. Find all natural numbers $p$ such that $S_{n,p}$ is a perfect square for each $n\in\mathbb N$.
2017 China National Olympiad, 6
Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$
1984 Czech And Slovak Olympiad IIIA, 3
Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation
$$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$
Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation
$$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$
where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1).
Note: $[x]$ indicates the whole part of the number $x$.
2001 National Olympiad First Round, 10
At each step, we are changing the places of exactly two numbers from the sequence $1$, $2$, $3$, $4$, $5$, $6$, $7$. How many different arrangements can be formed after two steps?
$
\textbf{(A)}\ 88
\qquad\textbf{(B)}\ 100
\qquad\textbf{(C)}\ 120
\qquad\textbf{(D)}\ 176
\qquad\textbf{(E)}\ 441
$
2018 AMC 12/AHSME, 17
Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $
1973 Spain Mathematical Olympiad, 2
Determine all solutions of the system
$$\begin{cases} 2x - 5y + 11z - 6 = 0 \\ -x + 3y - 16z + 8 = 0 \\ 4x - 5y - 83z + 38 = 0 \\ 3x + 11y - z + 9 > 0 \end{cases}$$
in which the first three are equations and the last one is a linear inequality.
2015 Saint Petersburg Mathematical Olympiad, 3
There are weights with mass $1,3,5,....,2i+1,...$ Let $A(n)$ -is number of different sets with total mass equal $n$( For example $A(9)=2$, because we have two sets $9=9=1+3+5$). Prove that $A(n) \leq A(n+1)$ for $n>1$
2003 IMO Shortlist, 8
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements;
(ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power.
What is the largest possible number of elements in $A$ ?
2023 Taiwan TST Round 1, A
Let $f:\mathbb{N}\to\mathbb{R}_{>0}$ be a given increasing function that takes positive values. For any pair $(m,n)$ of positive integers, we call it [i]disobedient[/i] if $f(mn)\neq f(m)f(n)$. For any positive integer $m$, we call it [i]ultra-disobedient[/i] if for any nonnegative integer $N$, there are always infinitely many positive integers $n$ satisfying that $(m,n), (m,n+1),\ldots,(m,n+N)$ are all disobedient pairs.
Show that if there exists some disobedient pair, then there exists some ultra-disobedient positive integer.
[i]
Proposed by usjl[/i]
Russian TST 2019, P3
Let $H{}$ be the orthocenter of the acute-angled triangle $ABC$. In the triangle $BHC$, the median $HM$ and the symedian $HL$ are drawn. The point $K{}$ is marked on the line $LH$ so that $\angle AKL=90^\circ$. Prove that the circumcircles of the triangles $ABC$ and $KLM$ are tangent.
2015 Middle European Mathematical Olympiad, 1
Prove that for all positive real numbers $a$, $b$, $c$ such that $abc=1$ the following inequality holds:
$$\frac{a}{2b+c^2}+\frac{b}{2c+a^2}+\frac{c}{2a+b^2}\le \frac{a^2+b^2+c^2}3.$$
1971 IMO Longlists, 25
Let $ABC,AA_1A_2,BB_1B_2, CC_1C_2$ be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments $A_2B_1,B_2C_1, C_2A_1$ by $P,Q,R$ in this order. Prove that the triangle $PQR$ is equilateral.
2020-21 KVS IOQM India, 29
Consider a permutation $(a_1,a_2,a_3,a_4,a_5)$ of $\{1,2,3,4,5\}$. We say the $5$-tuple $(a_1,a_2,a_3,a_4,a_5)$ is dlawless if for all $1 \le i<j<k \le 5$, the sequence $(a_i,a_j,a_k)$ is [b]not [/b] an arithmetic progression (in that order). Find the number of flawless $5$-tuples.
2003 AMC 8, 15
A fi gure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a fi gure with the front and side views shown?
[asy]
defaultpen(linewidth(0.8));
path p=unitsquare;
draw(p^^shift(0,1)*p^^shift(1,0)*p);
draw(shift(4,0)*p^^shift(5,0)*p^^shift(5,1)*p);
label("FRONT", (1,0), S);
label("SIDE", (5,0), S);[/asy]
$ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$
Brazil L2 Finals (OBM) - geometry, 2021.5
Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously.
a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint.
b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.
2012 ELMO Shortlist, 4
Circles $\Omega$ and $\omega$ are internally tangent at point $C$. Chord $AB$ of $\Omega$ is tangent to $\omega$ at $E$, where $E$ is the midpoint of $AB$. Another circle, $\omega_1$ is tangent to $\Omega, \omega,$ and $AB$ at $D,Z,$ and $F$ respectively. Rays $CD$ and $AB$ meet at $P$. If $M$ is the midpoint of major arc $AB$, show that $\tan \angle ZEP = \tfrac{PE}{CM}$.
[i]Ray Li.[/i]
1980 Austrian-Polish Competition, 8
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$
2012 AMC 10, 9
Two integers have a sum of $26$. When two more integers are added to the first two integers the sum is $41$. Finally when two more integers are added to the sum of the previous four integers the sum is $57$. What is the minimum number of even integers among the $6$ integers?
${{ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4}\qquad\textbf{(E)}\ 5} $