Found problems: 85335
1962 Poland - Second Round, 4
Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality
$$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality
$$ d_a > d_b > d_c.$$
1995 Tuymaada Olympiad, 2
Let $x_1=a, x_2=a^{x_1}, ..., x_n=a^{x_{n-1}}$ where $a>1$. What is the maximum value of $a$ for which lim exists $\lim_{n\to \infty} x_n$ and what is this limit?
2022 JHMT HS, 7
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does $2022$ of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
1955 AMC 12/AHSME, 39
If $ y\equal{}x^2\plus{}px\plus{}q$, then if the least possible value of $ y$ is zero $ q$ is equal to:
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \frac{p^2}{4} \qquad
\textbf{(C)}\ \frac{p}{2} \qquad
\textbf{(D)}\ \minus{}\frac{p}{2} \qquad
\textbf{(E)}\ \frac{p^2}{4}\minus{}q$
1989 IMO Shortlist, 5
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
2010 Contests, 3
If, instead, the graph is a graph of ACCELERATION vs. TIME and the squirrel starts from rest, then the squirrel has the greatest speed at what time(s) or during what time interval?
(A) at B
(B) at C
(C) at D
(D) at both B and D
(E) From C to D
2010 AMC 10, 2
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
[asy]unitsize(8mm);
defaultpen(linewidth(.8pt));
draw(scale(4)*unitsquare);
draw((0,3)--(4,3));
draw((1,3)--(1,4));
draw((2,3)--(2,4));
draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$
2015 India IMO Training Camp, 1
Let $n \ge 2$ be an integer, and let $A_n$ be the set \[A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.\] Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n$ .
[i]Proposed by Serbia[/i]
2009 India Regional Mathematical Olympiad, 3
Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.
2020 MIG, 14
Given that $x$ satisfies $2^{4x} \cdot 2^{4x} \cdot 8^{4x} = 16^5$, find the value of $x$.
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }10$
PEN R Problems, 11
Prove that if a lattice parallelogram contains at most three lattice points in addition to its vertices, then those are on one of the diagonals.
2023 BMT, Tie 3
Points $A$, $B$, and $C$ lie on a semicircle with diameter $\overline{PQ}$ such that $AB = 3$, $AC = 4$, $BC = 5$, and $A$ is on $\overline{PQ}$. Given $\angle PAB = \angle QAC$, compute the area of the semicircle.
2006 Germany Team Selection Test, 2
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
1985 Vietnam National Olympiad, 3
A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.
2014 AIME Problems, 4
Jon and Steve ride their bicycles on a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2005 iTest, 1
Joe finally asked Kathryn out. They go out on a date on a Friday night, racing at the local go-kart track. They take turns racing across an $8 \times 8$ square grid composed of $64$ unit squares. If Joe and Kathryn start in the lower left-hand corner of the $8\times 8$ square, and can move either up or right along any side of any unit square, what is the probability that Joe and Kathryn take the same exact path to reach the upper right-hand corner of the $8\times 8$ square grid?
2003 Flanders Junior Olympiad, 1
Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc.
Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal.
Who made it?
2013 Irish Math Olympiad, 9
We say that a doubly infinite sequence
$. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$
is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n.
(a) Find a subaveraging sequence in which all entries are different from each other. Prove that all
entries are indeed distinct.
(b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such
that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .
2024-25 IOQM India, 9
Consider the grid of points $X = \{(m,n) | 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $( c = a \pm 2$ and $d = b \pm 1)$ or $( c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:
2003 Swedish Mathematical Competition, 2
In a lecture hall some chairs are placed in rows and columns, forming a rectangle. In each row there are $6$ boys sitting and in each column there are $8$ girls sitting, whereas $15$ places are not taken. What can be said about the number of rows and that of columns?
1989 Tournament Of Towns, (231) 5
A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is
(a) less than $\frac14 MN$,
(b) less than $\frac15 MN$.
2017 Dutch IMO TST, 2
The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.
2018 AIME Problems, 3
Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
V Soros Olympiad 1998 - 99 (Russia), 10.2
Solve the equation $$ |\cos 3x - tgt| + |\cos 3x + tgt| = |tg^2t -3|.$$
2016 ITAMO, 2
A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ($0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are $7,1,2$ and $7,1,5$, but there might be two contestants whose ordered scores are $7,1,2$ and $7,2,1$). Find the maximum number of contestants.