Found problems: 95
2013 India IMO Training Camp, 3
We define an operation $\oplus$ on the set $\{0, 1\}$ by
\[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\]
For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.
For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.
2007 Bulgaria Team Selection Test, 3
Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.
2024 Korea Junior Math Olympiad (First Round), 13.
Find the number of positive integer n, which follows the following
$ \bigstar $ $ n=[\frac{m^3}{2024}] $ $n$ has a positive integer $m$ that follows this equation ($ m \le 1000$)
1998 Gauss, 19
Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point.
If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 11$
2001 India Regional Mathematical Olympiad, 2
Find all primes $p$ and $q$ such that $p^2 + 7pq + q^2$ is a perfect square.
1998 Gauss, 14
A cube has a volume of $125 ^3$ cm . What is the area of one face of the cube?
$\textbf{(A)}\ 20^2 \qquad \textbf{(B)}\ 25^2 \qquad \textbf{(C)}\ 41\frac{2}{3}^2 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 75$
2013 India IMO Training Camp, 3
We define an operation $\oplus$ on the set $\{0, 1\}$ by
\[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\]
For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$.
For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.
1999 Gauss, 5
Which one of the following gives an odd integer?
$\textbf{(A)}\ 6^2 \qquad \textbf{(B)}\ 23-17 \qquad \textbf{(C)}\ 9\times24 \qquad \textbf{(D)}\ 96\div8 \qquad \textbf{(E)}\ 9\times41$
1998 Gauss, 23
A cube measures $10 \text{cm} \times 10 \text{cm} \times10 \text{cm}$ . Three cuts are
made parallel to the faces of the cube as shown creating
eight separate solids which are then separated. What is the
increase in the total surface area?
$\textbf{(A)}\ 300 \text{cm}^2 \qquad \textbf{(B)}\ 800 \text{cm}^2 \qquad \textbf{(C)}\ 1200 \text{cm}^2 \qquad \textbf{(D)}\ 600 \text{cm}^2 \qquad \textbf{(E)}\ 0 \text{cm}^2$
1998 Gauss, 11
Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts were made
parallel to the sides of the original rectangle. In comparing R to S
(A) the area and perimeter both decrease
(B) the area decreases and the perimeter increases
(C) the area and perimeter both increase
(D) the area increases and the perimeter decreases
(E) the area decreases and the perimeter stays the same
1999 Gauss, 15
A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green?
$\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$
1998 Gauss, 22
Each time a bar of soap is used, its volume decreases by $10\%$.
What is the minimum number of times
a new bar would have to be used so that less than one-half its volume remains?
$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
2005 Romania National Olympiad, 1
Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$.
[i]Virgil Nicula[/i]
1999 Gauss, 4
$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}$ is equal to
$\textbf{(A)}\ \dfrac{15}{8} \qquad \textbf{(B)}\ 1\dfrac{3}{14} \qquad \textbf{(C)}\ \dfrac{11}{8} \qquad \textbf{(D)}\ 1\dfrac{3}{4} \qquad \textbf{(E)}\ \dfrac{7}{8}$
1989 China Team Selection Test, 3
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
1991 AMC 8, 18
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more?
[asy]
for(int a=1; a<11; ++a)
{
draw((a,0)--(a,-.5));
}
draw((0,10.5)--(0,0)--(10.5,0));
label("$1$",(1,-.5),S); label("$2$",(2,-.5),S); label("$3$",(3,-.5),S); label("$4$",(4,-.5),S);
label("$5$",(5,-.5),S); label("$6$",(6,-.5),S); label("$7$",(7,-.5),S); label("$8$",(8,-.5),S);
label("$9$",(9,-.5),S); label("$10$",(10,-.5),S); label("Number of years with company",(5.5,-2),S);
label("X",(1,0),N); label("X",(1,1),N); label("X",(1,2),N); label("X",(1,3),N); label("X",(1,4),N);
label("X",(2,0),N); label("X",(2,1),N); label("X",(2,2),N); label("X",(2,3),N); label("X",(2,4),N);
label("X",(3,0),N); label("X",(3,1),N); label("X",(3,2),N); label("X",(3,3),N);
label("X",(3,4),N); label("X",(3,5),N); label("X",(3,6),N); label("X",(3,7),N);
label("X",(4,0),N); label("X",(4,1),N); label("X",(4,2),N); label("X",(5,0),N); label("X",(5,1),N);
label("X",(6,0),N); label("X",(6,1),N); label("X",(7,0),N); label("X",(7,1),N);
label("X",(8,0),N); label("X",(9,0),N); label("X",(10,0),N);
label("Gauss Company",(5.5,10),N);
[/asy]
$\text{(A)}\ 9\% \qquad \text{(B)}\ 23\frac{1}{3}\% \qquad \text{(C)}\ 30\% \qquad \text{(D)}\ 42\frac{6}{7}\% \qquad \text{(E)}\ 50\% $
1998 Gauss, 17
Claire takes a square piece of paper and folds it in half four times without unfolding, making an
isosceles right triangle each time. After unfolding the paper to form a square again, the creases on the
paper would look like
PEN C Problems, 2
The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
2006 Stanford Mathematics Tournament, 19
When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from 1 through 100. He quickly added 1 and 100, 2 and 99, and so on for 50 pairs of numbers each adding in 101. His answer was 50 · 101=5,050. Now find the sum of all the digits in the integers from 1 through 1,000,000 (i.e. all the digits in those numbers, not the numbers themselves).
1999 Gauss, 22
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$