Found problems: 85335
1981 Brazil National Olympiad, 2
Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?
2021 Czech-Polish-Slovak Junior Match, 1
You are given a $2 \times 2$ array with a positive integer in each field. If we add the product of the numbers in the first column, the product of the numbers in the second column, the product of the numbers in the first row and the product of the numbers in the second row, we get $2021$.
a) Find possible values for the sum of the four numbers in the table.
b) Find the number of distinct arrays that satisfy the given conditions that contain four pairwise distinct numbers in arrays.
LMT Team Rounds 2021+, 3
Adamand Topher are playing a game in which each of them starts with $2$ pickles. Each turn, they flip a fair coin: if it lands heads, Topher takes $1$ pickle from Adam; if it lands tails, Adam takes $2$ pickles from Topher. (If Topher has only $1$ pickle left, Adam will just take it.) What’s the probability that Topher will have all $4$ pickles before Adam does?
1990 Baltic Way, 14
Do there exist $1990$ pairwise coprime positive integers such that all sums of two or more of these numbers are composite numbers?
2019 Korea Winter Program Practice Test, 2
$\omega_1,\omega_2$ are orthogonal circles, and their intersections are $P,P'$. Another circle $\omega_3$ meets $\omega_1$ at $Q,Q'$, and $\omega_2$ at $R,R'$. (The points $Q,R,Q',R'$ are in clockwise order.) Suppose $P'R$ and $PR'$ meet at $S$, and let $T$ be the circumcenter of $\triangle PQR$. Prove that $T,Q,S$ are collinear if and only if $O_1,S,O_3$ are collinear. ($O_i$ is the center of $\omega_i$ for $i=1,2,3$.)
1975 IMO, 5
Can there be drawn on a circle of radius $1$ a number of $1975$ distinct points, so that the distance (measured on the chord) between any two points (from the considered points) is a rational number?
2012 Finnish National High School Mathematics Competition, 1
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
Russian TST 2021, P3
Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where $m$ is a positive integer and $q_1,\ldots , q_m$ are polynomials with integer coefficients, and $p_1,\ldots , p_m$ are good polynomials. Find the smallest natural number $D{}$ such that each monomial of degree $D{}$ lies in the set $\mathcal{J}$.
PEN A Problems, 111
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.
2010 Philippine MO, 2
On a cyclic quadrilateral $ABCD$, there is a point $P$ on side $AD$ such that the triangle $CDP$ and the quadrilateral $ABCP$ have equal perimeters and equal areas. Prove that two sides of $ABCD$ have equal lengths.
1939 Moscow Mathematical Olympiad, 049
Let the product of two polynomials of a variable $x$ with integer coefficients be a polynomial with even coefficients not all of which are divisible by $4$. Prove that all the coefficients of one of the polynomials are even and that at least one of the coefficients of the other polynomial is odd.
2001 Manhattan Mathematical Olympiad, 2
There are $2001$ marked points in the plane. It is known that the area of any triangle with vertices at the given points is less than or equal than $1 \ cm^2$. Prove that there exists a triangle with area no more than $4 \ cm^2$, which contains all $2001$ points.
Denmark (Mohr) - geometry, 1993.4
In triangle $ABC$, points $D, E$, and $F$ intersect one-third of the respective sides.
Show that the sum of the areas of the three gray triangles is equal to the area of middle triangle.
[img]https://1.bp.blogspot.com/-KWrhwMHXfDk/XzcIkhWnK5I/AAAAAAAAMYk/Tj6-PnvTy9ksHgke8cDlAjsj2u421Xa9QCLcBGAsYHQ/s0/1993%2BMohr%2Bp4.png[/img]
Estonia Open Senior - geometry, 2002.1.4
In a triangle $ABC$ we have $\angle B = 2 \cdot \angle C$ and the angle bisector drawn from $A$ intersects $BC$ in a point $D$ such that $|AB| = |CD|$. Find $\angle A$.
2015 Princeton University Math Competition, A2/B3
What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?
2017-IMOC, C4
There are $3N+1$ students with different heights line up for asking questions. Prove that the teacher can drive $2N$ students away such that the remain students satisfies: No one has neighbors whose heights are consecutive.
2005 Purple Comet Problems, 11
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
1981 National High School Mathematics League, 7
The equation $x|x|+px+q=0$ is given. Which of the following is not true?
$\text{(A)}$It has at most three real roots.
$\text{(B)}$It has at least one real root.
$\text{(C)}$Only if $p^2-4q\geq0 $,it has real roots.
$\text{(D)}$If $p<0$ and $q>0$, it has three real roots.
2014 Contests, 3
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.
(Day 1, 3rd problem
author: Michal Rolínek)
Kvant 2024, M2803
Given is a permutation of $1, 2, \ldots, 2023, 2024$ and two positive integers $a, b$, such that for any two adjacent numbers, at least one of the following conditions hold:
1) their sum is $a$;
2) the absolute value of their difference is $b$.
Find all possible values of $b$.
Kvant 2019, M2567
On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.
2004 Iran Team Selection Test, 2
Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.
Estonia Open Junior - geometry, 2019.1.5
Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$. Lines $AM$ and $BC$ intersect in point $N$. What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$?
2017 ASDAN Math Tournament, 2
An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap).
2021 Indonesia TST, N
For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.